{"id":888,"date":"2026-03-11T11:15:55","date_gmt":"2026-03-11T10:15:55","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=888"},"modified":"2026-03-17T06:39:22","modified_gmt":"2026-03-17T05:39:22","slug":"mad-ecuaciones-diofanticas-y-de-congruencias-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=888","title":{"rendered":"MAD: Ecuaciones diof\u00e1nticas y de congruencias con maxima"},"content":{"rendered":"<h2>Ecuaci\u00f3n de congruencias<\/h2>\n<p>Recordemos, para resolver la ecuaci\u00f3n  \\(aX\\equiv b {\\pmod {n}}\\), \\((aX \\equiv b (n))\\), cuando \\(\\mathbf{mcd}(a,n)=1\\), utilizamos bien soluci\u00f3n de B\u00e9zout, bien la funci\u00f3n \\(\\varphi\\) de Euler.<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Resolver la ecuaci\u00f3n \\[7X\\equiv 3 {\\pmod {103}}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2d3() {\n  var htmlShow2d3 = document.getElementById(\"html-show2d3\");\n  if (htmlShow2d3.style.display === \"none\") {\n    htmlShow2d3.style.display = \"block\";\n  } else {\n    htmlShow2d3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2d3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d3\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">ex<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span>(<span class=\"code_number\">7<\/span>,<span class=\"code_number\">103<\/span>)<span class=\"code_endofline\">;<\/span><br \/>if <span class=\"code_variable\">ex<\/span>[<span class=\"code_number\">1<\/span>]&lt;<span class=\"code_number\">0<\/span> then <span class=\"code_variable\">inv<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">ex<\/span>[<span class=\"code_number\">1<\/span>],<span class=\"code_number\">103<\/span>) else <span class=\"code_variable\">inv<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">ex<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">sol<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">inv<\/span><span class=\"code_operator\">*<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">103<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abSoluci\u00f3n X=\u00bb<\/span>,<span class=\"code_variable\">sol<\/span>,<span class=\"code_string\">\u00ab(103)\u00bb<\/span>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(ex)<\/mtext><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mn>44<\/mn><mo>,<\/mo><mn>3<\/mn><mo>,<\/mo><mn>1<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>Soluci\u00f3n X=<\/mo><mo\/><mn>74<\/mn><mo\/><mo>(103)<\/mo><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Verifiqu\u00e9moslo:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">mod<\/span>(<span class=\"code_number\">7<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">sol<\/span>,<span class=\"code_number\">103<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o6) <\/mtext><\/mtd><mtd><mn>3<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Resolver la ecuaci\u00f3n de congruencias \\(3X\\equiv 16 \\pmod{22}\\).<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5w() {\n  var htmlShow5w = document.getElementById(\"html-show5w\");\n  if (htmlShow5w.style.display === \"none\") {\n    htmlShow5w.style.display = \"block\";\n  } else {\n    htmlShow5w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv5w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5w\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">block<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">id<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">for <\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span><span class=\"code_function\"> while <\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">&gt;<\/span><span class=\"code_number\">0<\/span><span class=\"code_function\"> do <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">r<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --> <\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">fiEuler<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">block<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">f<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0 <span class=\"code_variable\">f<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0 <span class=\"code_function\">for <\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span><span class=\"code_function\"> thru <\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_function\"> do <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">if<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> then <\/span><span class=\"code_variable\">f<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">f<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span>\u00a0 <span class=\"code_variable\">f<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">fi<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">fiEuler<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">22<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }10\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Sabemos que \\(3^{10-1}\\pmod{22}\\) es el inverso de 3, luego<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">22<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">22<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">9<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\left[ 5\\operatorname{,}15\\operatorname{,}1\\operatorname{,}3\\operatorname{,}9\\operatorname{,}5\\operatorname{,}15\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El inverso que buscamos es 15. El resultado ser\u00e1<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">15<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_number\">16<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">22<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }20\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Verifiqu\u00e9moslo<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n<span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_number\">20<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">22<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }16\\]<\/p>\n<\/div>\n<hr \/>\n<p>Hemos visto el uso de la funci\u00f3n \\(\\varphi\\) de Euler. El c\u00e1lculo de esta funci\u00f3n tambi\u00e9n se puede realizar si sabemos la descomposici\u00f3n de un n\u00famero entero en sus factores primos:<\/p>\n<blockquote>\n<p>Si \\(n\\in\\mathbb{Z}\\) es \\({\\displaystyle n=p_{1}^{k_{1}}\\cdots p_{r}^{k_{r}}}\\), entonces \\[\\varphi (n)=n\\prod_{i=1}^r\\left(1-\\frac{1}{p_i}\\right)\\]<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Utilizar la f\u00f3rmula anterior para construir un algoritmo que calcule \\(\\varphi(87)\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2d8() {\n  var htmlShow2d8 = document.getElementById(\"html-show2d8\");\n  if (htmlShow2d8.style.display === \"none\") {\n    htmlShow2d8.style.display = \"block\";\n  } else {\n    htmlShow2d8.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2d8()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d8\" style=\"display: none;\">\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i2)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">fiEuler<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span> <span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span> <span class=\"code_function\">block<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">result<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_variable\">p<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_variable\">d<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_variable\">result<\/span> <span class=\"code_operator\">:<\/span> <span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_variable\">p<\/span> <span class=\"code_operator\">:<\/span> <span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_variable\">d<\/span> <span class=\"code_operator\">:<\/span> <span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span> \u00a0 <span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_function\">while<\/span> <span class=\"code_variable\">d<\/span> <span class=\"code_operator\">\u00b7<\/span> <span class=\"code_variable\">d<\/span> <span class=\"code_endofline\">&lt;<\/span><span class=\"code_operator\">=<\/span> <span class=\"code_variable\">p<\/span> <span class=\"code_function\">do<\/span> <span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">if<\/span> <span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_variable\">d<\/span><span class=\"code_operator\">)<\/span> <span class=\"code_operator\">=<\/span> <span class=\"code_number\">0<\/span> <span class=\"code_function\">then<\/span> <span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_comment\">\/* Aplicamos la f\u00f3rmula: result = result * (1 &#8211; 1\/d) *\/<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">result<\/span> <span class=\"code_operator\">:<\/span> <span class=\"code_variable\">result<\/span> <span class=\"code_operator\">\u00b7<\/span> <span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span> <span class=\"code_operator\">\u2212<\/span> <span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span> <span class=\"code_operator\">\/<\/span> <span class=\"code_variable\">d<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_comment\">\/* Eliminamos todas las ocurrencias del factor d en p *\/<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">while<\/span> <span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_variable\">d<\/span><span class=\"code_operator\">)<\/span> <span class=\"code_operator\">=<\/span> <span class=\"code_number\">0<\/span> <span class=\"code_function\">do<\/span> <span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">p<\/span> <span class=\"code_operator\">:<\/span> <span class=\"code_variable\">p<\/span> <span class=\"code_operator\">\/<\/span> <span class=\"code_variable\">d<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">d<\/span> <span class=\"code_operator\">:<\/span> <span class=\"code_variable\">d<\/span> <span class=\"code_operator\">+<\/span> <span class=\"code_number\">1<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_comment\">\/* Si al final p &gt; 1, lo que queda es el \u00faltimo factor primo *\/<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_function\">if<\/span> <span class=\"code_variable\">p<\/span> <span class=\"code_endofline\">&gt;<\/span> <span class=\"code_number\">1<\/span> <span class=\"code_function\">then<\/span> <span class=\"code_variable\">result<\/span> <span class=\"code_operator\">:<\/span> <span class=\"code_variable\">result<\/span> <span class=\"code_operator\">\u00b7<\/span> <span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/span> <span class=\"code_operator\">\u2212<\/span> <span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span> <span class=\"code_operator\">\/<\/span> <span class=\"code_variable\">p<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0 <span class=\"code_function\">return<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">result<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">fiEuler<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">87<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\tag{%o2} 56\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Sistemas de congruencias <\/h2>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Resolver el sistema \\[\\begin{align*}<br \/>\n3X &#038;\\equiv 2\\pmod{8} \\\\  2X&#038;\\equiv 1\\pmod{5} \\end{align*}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1b() {\n  var htmlShow1b = document.getElementById(\"html-show1b\");\n  if (htmlShow1b.style.display === \"none\") {\n    htmlShow1b.style.display = \"block\";\n  } else {\n    htmlShow1b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1b\" style=\"display: none;\">\n<div class=\"comment\">\n<p>Tenemos las ecuaciones de congruencia 3X\\(\\equiv\\)2(mod 8) y 2X\\(\\equiv\\)1(mod 5):<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span>[[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">8<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">5<\/span>]]<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Resolvamos la ecuaciones de congruencia de forma individual<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span>[]<span class=\"code_endofline\">$<\/span><br \/>for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span> thru <span class=\"code_number\">2<\/span> do(<br \/> \u00a0\u00a0 <span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span>(<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>]),<br \/> \u00a0\u00a0 <span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span>(<span class=\"code_variable\">s<\/span>,[<span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">d<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">2<\/span>],<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>])]),<br \/> \u00a0\u00a0 <span class=\"code_function\">print<\/span>(<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">1<\/span>],<span class=\"code_string\">\u00abX=\u00bb<\/span>,<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">2<\/span>],<span class=\"code_string\">\u00ab(mod\u00bb<\/span>,<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_string\">\u00ab)-&gt; X=\u00bb<\/span>,<span class=\"code_variable\">s<\/span>[<span class=\"code_variable\">i<\/span>],<span class=\"code_string\">\u00ab(mod\u00bb<\/span>,<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_string\">\u00ab)\u00bb<\/span>)<br \/>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mn>3<\/mn><mo\/><mo>X=<\/mo><mo\/><mn>2<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>8<\/mn><mo\/><mo>)\u2192 X=<\/mo><mo\/><mn>6<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>8<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mn>2<\/mn><mo\/><mo>X=<\/mo><mo\/><mn>1<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>5<\/mn><mo\/><mo>)\u2192 X=<\/mo><mo\/><mn>3<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>5<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Ahora calculamos los inversos de los respectivos m\u00f3dulos m\u00f3dulo el producto de ellos:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">sv<\/span><span class=\"code_operator\">:<\/span>[]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">p<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">product<\/span>(<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">$<\/span><br \/>for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span> thru <span class=\"code_number\">2<\/span> do(<br \/> \u00a0\u00a0 <span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span>(<span class=\"code_variable\">p<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>]),<br \/> \u00a0\u00a0 <span class=\"code_variable\">sv<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span>(<span class=\"code_variable\">sv<\/span>,[<span class=\"code_variable\">d<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">p<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>]])<br \/>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">sv<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o10) <\/mtext><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mn>15<\/mn><mo>,<\/mo><mn>16<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>La soluci\u00f3n del sistema ser\u00e1<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i13) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">sm<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">$<\/span><br \/>for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span> thru <span class=\"code_number\">2<\/span> do(<br \/> \u00a0\u00a0 <span class=\"code_variable\">sm<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">sm<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">s<\/span>[<span class=\"code_variable\">i<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">sv<\/span>[<span class=\"code_variable\">i<\/span>]<br \/>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abX=\u00bb<\/span>,<span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">sm<\/span>,<span class=\"code_variable\">p<\/span>),<span class=\"code_string\">\u00ab(mod\u00bb<\/span>,<span class=\"code_variable\">p<\/span>, <span class=\"code_string\">\u00ab)\u00bb<\/span>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>X=<\/mo><mo\/><mn>38<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>40<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math><\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Resolver el sistema \\begin{align*}<br \/>\n 5&#038;\\equiv 3 \\pmod{17} \\\\<br \/>\n 2&#038;\\equiv 7 \\pmod{15} \\\\<br \/>\n 12&#038;\\equiv 14\\pmod{31} \\\\<br \/>\n 22&#038;\\equiv 17\\pmod{29} \\\\<br \/>\n\\end{align*}  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1a() {\n  var htmlShow1a = document.getElementById(\"html-show1a\");\n  if (htmlShow1a.style.display === \"none\") {\n    htmlShow1a.style.display = \"block\";\n  } else {\n    htmlShow1a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1a\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Tenemos la ecuaciones de congruencia 5X=3(mod 17),2X=7(mod 15), 12X=14(mod 31) y22X=17(mod 29):<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span>[[<span class=\"code_number\">5<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">17<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">7<\/span>,<span class=\"code_number\">15<\/span>],[<span class=\"code_number\">12<\/span>,<span class=\"code_number\">14<\/span>,<span class=\"code_number\">31<\/span>],[<span class=\"code_number\">22<\/span>,<span class=\"code_number\">17<\/span>,<span class=\"code_number\">29<\/span>]]<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Resolvamos la ecuaciones de congruencia de forma individual<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span>[]<span class=\"code_endofline\">$<\/span><br \/>for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span> thru <span class=\"code_function\">length<\/span>(<span class=\"code_variable\">eq<\/span>) do(<br \/> \u00a0\u00a0 <span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span>(<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>]),<br \/> \u00a0\u00a0 <span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span>(<span class=\"code_variable\">s<\/span>,[<span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">d<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">2<\/span>],<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>])]),<br \/> \u00a0\u00a0 <span class=\"code_function\">print<\/span>(<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">1<\/span>],<span class=\"code_string\">\u00abX=\u00bb<\/span>,<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">2<\/span>],<span class=\"code_string\">\u00ab(mod\u00bb<\/span>,<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_string\">\u00ab)-&gt; X=\u00bb<\/span>,<span class=\"code_variable\">s<\/span>[<span class=\"code_variable\">i<\/span>],<span class=\"code_string\">\u00ab(mod\u00bb<\/span>,<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_string\">\u00ab)\u00bb<\/span>)<br \/>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mn>5<\/mn><mo\/><mo>X=<\/mo><mo\/><mn>3<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>17<\/mn><mo\/><mo>)\u2192 X=<\/mo><mo\/><mn>4<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>17<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mn>2<\/mn><mo\/><mo>X=<\/mo><mo\/><mn>7<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>15<\/mn><mo\/><mo>)\u2192 X=<\/mo><mo\/><mn>11<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>15<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mn>12<\/mn><mo\/><mo>X=<\/mo><mo\/><mn>14<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>31<\/mn><mo\/><mo>)\u2192 X=<\/mo><mo\/><mn>27<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>31<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mn>22<\/mn><mo\/><mo>X=<\/mo><mo\/><mn>17<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>29<\/mn><mo\/><mo>)\u2192 X=<\/mo><mo\/><mn>10<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>29<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Ahora calculamos los inversos de los respectivos m\u00f3dulos m\u00f3dulo el producto de ellos:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">sv<\/span><span class=\"code_operator\">:<\/span>[]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">p<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">product<\/span>(<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_function\">length<\/span>(<span class=\"code_variable\">eq<\/span>))<span class=\"code_endofline\">$<\/span><br \/>for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span> thru <span class=\"code_function\">length<\/span>(<span class=\"code_variable\">eq<\/span>) do(<br \/> \u00a0\u00a0 <span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span>(<span class=\"code_variable\">p<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>],<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>]),<br \/> \u00a0\u00a0 <span class=\"code_variable\">sv<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span>(<span class=\"code_variable\">sv<\/span>,[<span class=\"code_variable\">d<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">p<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>]])<br \/>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">sv<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o10) <\/mtext><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mn>53940<\/mn><mo>,<\/mo><mn>106981<\/mn><mo>,<\/mo><mn>81345<\/mn><mo>,<\/mo><mn>94860<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>La soluci\u00f3n del sistema ser\u00e1<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i14) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">sm<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">$<\/span><br \/>for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span> thru <span class=\"code_function\">length<\/span>(<span class=\"code_variable\">eq<\/span>) do(<br \/> \u00a0\u00a0 <span class=\"code_variable\">sm<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">sm<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">s<\/span>[<span class=\"code_variable\">i<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">sv<\/span>[<span class=\"code_variable\">i<\/span>]<br \/>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">sol<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">sm<\/span>,<span class=\"code_variable\">p<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abX=\u00bb<\/span>,<span class=\"code_variable\">sol<\/span>,<span class=\"code_string\">\u00ab(mod\u00bb<\/span>,<span class=\"code_variable\">p<\/span>, <span class=\"code_string\">\u00ab)\u00bb<\/span>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>X=<\/mo><mo\/><mn>208781<\/mn><mo\/><mo>(mod<\/mo><mo\/><mn>229245<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Veamos que es cierto:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i15) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">makelist<\/span>(<span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">sol<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">2<\/span>],<span class=\"code_variable\">eq<\/span>[<span class=\"code_variable\">i<\/span>][<span class=\"code_number\">3<\/span>]),<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_function\">length<\/span>(<span class=\"code_variable\">eq<\/span>))<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o15) <\/mtext><\/mtd><mtd><mo>[<\/mo><mn>0<\/mn><mo>,<\/mo><mn>0<\/mn><mo>,<\/mo><mn>0<\/mn><mo>,<\/mo><mn>0<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<h2>Ecuaci\u00f3n lineal diof\u00e1ntica<\/h2>\n<p>El pasado d\u00eda trabajamos las ecuaciones lineales diof\u00e1nticas. En particular, abordamos la soluci\u00f3n de la ecuaci\u00f3n \\[ax+by=c.\\]<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(v\\) la menor de las soluciones positivas de 4x+22y=46, \u00bfcu\u00e1l es el valor de [3,-5].v?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1d() {\n  var htmlShow1d = document.getElementById(\"html-show1d\");\n  if (htmlShow1d.style.display === \"none\") {\n    htmlShow1d.style.display = \"block\";\n  } else {\n    htmlShow1d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1d\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Utilicemos el algoritmo extendido de Euclides para determinar el mcd y la soluci\u00f3n de B\u00e9zout:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span>(<span class=\"code_number\">4<\/span>,<span class=\"code_number\">22<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(d)<\/mtext><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mn>5<\/mn><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Veamos si verifica las condiciones y, en su caso, sustiyamos en la f\u00f3rmula:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\">if(<span class=\"code_function\">mod<\/span>(<span class=\"code_number\">46<\/span>,<span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>])<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span>) then (<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abHay soluci\u00f3n:\u00bb<\/span>),<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">46<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>],<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">x<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">22<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">k<\/span>,<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">y<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">2<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">k<\/span>,<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abx=\u00bb<\/span>,<span class=\"code_variable\">x<\/span>),<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00aby=\u00bb<\/span>,<span class=\"code_variable\">y<\/span>)<br \/> \u00a0\u00a0 )<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>Hay soluci\u00f3n:<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>x=<\/mo><mo\/><mn>11<\/mn><mo>\u2062<\/mo><mi>k<\/mi><mi>\u2212<\/mi><mn>115<\/mn><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>y=<\/mo><mo\/><mn>23<\/mn><mi>\u2212<\/mi><mn>2<\/mn><mo>\u2062<\/mo><mi>k<\/mi><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Estimemos las soluciones positivas. Primero creamos la funci\u00f3n que nos devuelve las soluciones:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">sol<\/span>(<span class=\"code_variable\">i<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span>[<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span>),<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span>)]<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Ahora, consideramos el valor m\u00e1s cercano de la x a cero:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"> <span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">id<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">floor<\/span>(<span class=\"code_number\">115<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">11<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">makelist<\/span>(<span class=\"code_function\">sol<\/span>(<span class=\"code_variable\">i<\/span>),<span class=\"code_variable\">i<\/span>,<span class=\"code_variable\">id<\/span>,<span class=\"code_variable\">id<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o6) <\/mtext><\/mtd><mtd><mo>[<\/mo><mo>[<\/mo><mi>\u2212<\/mi><mn>5<\/mn><mo>,<\/mo><mn>3<\/mn><mo>]<\/mo><mo>,<\/mo><mo>[<\/mo><mn>6<\/mn><mo>,<\/mo><mn>1<\/mn><mo>]<\/mo><mo>,<\/mo><mo>[<\/mo><mn>17<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>1<\/mn><mo>]<\/mo><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Respondamos a la pregunta:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\">[<span class=\"code_number\">6<\/span>,<span class=\"code_number\">1<\/span>].[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">5<\/span>]<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o7) <\/mtext><\/mtd><mtd><mn>13<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(v\\) la menor de las soluciones positivas de 7x-11y=5, \u00bfcu\u00e1l es el valor de [3,4].v?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2d() {\n  var htmlShow2d = document.getElementById(\"html-show2d\");\n  if (htmlShow2d.style.display === \"none\") {\n    htmlShow2d.style.display = \"block\";\n  } else {\n    htmlShow2d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Como necesitamos el mcd y, si hay soluci\u00f3n, una soluci\u00f3n inicial dada por la soluci\u00f3n de Bezout, apliquemos el algoritmo extendido de Euclides<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_variable\">a<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">11<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span>(<span class=\"code_variable\">a<\/span>,<span class=\"code_variable\">b<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(d)<\/mtext><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mn>3<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>2<\/mn><mo>,<\/mo><mn>1<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>El mcd es 1, con lo cual hay soluci\u00f3n. Apliquemos la f\u00f3rmula:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">sol<\/span>(<span class=\"code_variable\">k<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span>[<span class=\"code_variable\">d<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">*<\/span>(<span class=\"code_variable\">c<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>])<span class=\"code_operator\">+<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">k<\/span>,<span class=\"code_variable\">d<\/span>[<span class=\"code_number\">2<\/span>]<span class=\"code_operator\">*<\/span>(<span class=\"code_variable\">c<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>])<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span>[<span class=\"code_number\">3<\/span>]<span class=\"code_operator\">*<\/span><span class=\"code_variable\">k<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">sol<\/span>(<span class=\"code_variable\">k<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o8) <\/mtext><\/mtd><mtd><mo>[<\/mo><mi>\u2212<\/mi><mn>11<\/mn><mo>\u2062<\/mo><mi>k<\/mi><mi>\u2212<\/mi><mn>15<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>7<\/mn><mo>\u2062<\/mo><mi>k<\/mi><mi>\u2212<\/mi><mn>10<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Buscamos una soluci\u00f3n positiva, luego<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i11) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">solve<\/span>(<span class=\"code_function\">sol<\/span>(<span class=\"code_variable\">k<\/span>)[<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">k<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_variable\">p<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">floor<\/span>(<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">k<\/span>,<span class=\"code_variable\">%<\/span>))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">makelist<\/span>(<span class=\"code_function\">sol<\/span>(<span class=\"code_variable\">i<\/span>),<span class=\"code_variable\">i<\/span>,<span class=\"code_variable\">p<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_variable\">p<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o9) <\/mtext><\/mtd><mtd><mo>[<\/mo><mi>k<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mfrac><mn>15<\/mn><mn>11<\/mn><\/mfrac><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(p)<\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>2<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o11) <\/mtext><\/mtd><mtd><mo>[<\/mo><mo>[<\/mo><mn>29<\/mn><mo>,<\/mo><mn>18<\/mn><mo>]<\/mo><mo>,<\/mo><mo>[<\/mo><mn>18<\/mn><mo>,<\/mo><mn>11<\/mn><mo>]<\/mo><mo>,<\/mo><mo>[<\/mo><mn>7<\/mn><mo>,<\/mo><mn>4<\/mn><mo>]<\/mo><mo>,<\/mo><mo>[<\/mo><mi>\u2212<\/mi><mn>4<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>3<\/mn><mo>]<\/mo><mo>,<\/mo><mo>[<\/mo><mi>\u2212<\/mi><mn>15<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>10<\/mn><mo>]<\/mo><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Como vemos, k=-2 es el n\u00famero que marca el cambio de signo en la variable x, y en este caso tambien en la y. Por tanto, la soluci\u00f3n que buscamos es<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">sol<\/span>(<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>).[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">4<\/span>]<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o12) <\/mtext><\/mtd><mtd><mn>37<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr>\n<p>Si la ecuaci\u00f3n es de tres variables y tiene soluci\u00f3n, el caso m\u00e1s sencillo es cuando tenemos dos coeficientes coprimos.  En tal caso, la ecuaci\u00f3n plantea una soluci\u00f3n param\u00e9trica cuyo par\u00e1metro es la variable del coeficiente no coprimo. Es decir, si \\(m.c.d(a,b)=1\\), planteamos la ecuaci\u00f3n \\[ax+by=n-c\\lambda,\\] donde designamos \\(z=\\lambda\\), la resolvemos como ya conocemos.<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Resolver la ecuaci\u00f3n \\(5x+12y+3z=11\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3d() {\n  var htmlShow3d = document.getElementById(\"html-show3d\");\n  if (htmlShow3d.style.display === \"none\") {\n    htmlShow3d.style.display = \"block\";\n  } else {\n    htmlShow3d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3d\" style=\"display: none;\">\n<div class=\"comment\">\nUtilicemos el algoritmo de Euclides para determinar el mcd\n<\/div>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i6)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">12<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">11<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o6} \\left[ 1\\mathop{,}1\\right] \\]\n<\/p>\n<div class=\"comment\">\nComo el mcd es 1 la ecuaci\u00f3n tendr\u00e1 soluci\u00f3n. Ahora elegimos dos coeficientes coprimos: ve\u00edamos que el 5 y el 12 los son. Resolvemos, por tanto, la ecuaci\u00f3n 5x+12y=11-3k, considerando z=k\n<\/div>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i7)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">12<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o7} \\left[ 5\\mathop{,}\\mathop{-}2\\mathop{,}1\\right] \\]\n<\/p>\n<div class=\"comment\">\nObservemos que la Soluci\u00f3n de Bezout nos dice: 5*5+12*(-2)=1, luego\n<\/div>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i10)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">11<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">12<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o9)\/R} 12 l\\mathop{-}15 k\\mathop{+}55\\]\n<\/p>\n<p>\n\\[\\]\\[\\tag{%o10)\/R} \\mathop{-}\\left( 5 l\\right) \\mathop{+}6 k\\mathop{-}22\\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nLas soluciones dependen de dos par\u00e1metros:\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i11)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nVeamos unas primeras soluciones:\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i12)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">create_list<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\begin{align*}[&#038;[55,-22,0],[67,-27,0],[79,-32,0],\\\\ &#038;[40,-16,1],[52,-21,1],[64,-26,1],\\\\ &#038;[25,-10,2],[37,-15,2],[49,-20,2]]\\end{align*}\\]\\[\\tag{%o12)\/R} \\]\n<\/p>\n<\/div>\n<hr>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la ecuaci\u00f3n 7x-2y+9z=10, dar la soluci\u00f3n positiva con min{x>0} y min{y>0}<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4a() {\n  var htmlShow4a = document.getElementById(\"html-show4a\");\n  if (htmlShow4a.style.display === \"none\") {\n    htmlShow4a.style.display = \"block\";\n  } else {\n    htmlShow4a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4a\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i6)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">9<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">10<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o6} \\left[ 2\\mathop{,}1\\right] \\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nVemos que tiene soluci\u00f3n, adem\u00e1s los coficientes son coprimos; por tanto, podemos elegir cualquier para de coeficientes para buscar la soluci\u00f3n. Sin embargo, en este caso nos piden una soluci\u00f3n que cumpla min{x&gt;0} y min{y&gt;0}, entonces nos interesa que x dependa de un \u00fanico par\u00e1metro. Es decir, resolvamos la ecuaci\u00f3n  -2y+9z=10-7k, considerando x=k:\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i7)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o7} \\left[ 4\\mathop{,}1\\mathop{,}1\\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i12)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o12)\/R} \\left[ k\\mathop{,}9 l\\mathop{-}28 k\\mathop{+}40\\mathop{,}2 l\\mathop{-}7 k\\mathop{+}10\\right] \\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nPara valores  min{x&gt;0}, ser\u00e1 x=1, con lo cual\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i13)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">x1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o13)\/R} \\left[ 1\\mathop{,}9 l\\mathop{+}12\\mathop{,}2 l\\mathop{+}3\\right] \\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nVeamos valores de y para estos puntos:\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i14)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o14)\/R} \\left[ \\left[ 1\\mathop{,}\\mathop{-}6\\mathop{,}\\mathop{-}1\\right] \\mathop{,}\\left[ 1\\mathop{,}3\\mathop{,}1\\right] \\mathop{,}\\left[ 1\\mathop{,}12\\mathop{,}3\\right] \\mathop{,}\\left[ 1\\mathop{,}21\\mathop{,}5\\right] \\mathop{,}\\left[ 1\\mathop{,}30\\mathop{,}7\\right] \\right] \\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nRepitamos el proceso considerando primero que y=k:\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i20)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o20)\/R} \\left[ 9 l\\mathop{+}48\\mathop{,}1\\mathop{,}\\mathop{-}\\left( 7 l\\right) \\mathop{-}36\\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i21)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o21)\/R} \\left[ \\left[ \\mathop{-}15\\mathop{,}1\\mathop{,}13\\right] \\mathop{,}\\left[ \\mathop{-}6\\mathop{,}1\\mathop{,}6\\right] \\mathop{,}\\left[ 3\\mathop{,}1\\mathop{,}\\mathop{-}1\\right] \\mathop{,}\\left[ 12\\mathop{,}1\\mathop{,}\\mathop{-}8\\right] \\right] \\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nComo la soluci\u00f3n [3,1,-1] no es positiva, la respuesta a la pregunta ser\u00eda la soluci\u00f3n [1,3,1].\n<\/div>\n<\/div>\n<hr>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea <strong>v<\/strong>\\(:[x_s,y_s,z_s]\\) la soluci\u00f3n de la ecuaci\u00f3n 5x-15y+22z=32, que cumple min{0<\\(y_s\\)<5} y max{0<\\(z_s\\)<10}, \u00bfcu\u00e1l es el producto de <strong>v<\/strong>.[-2,3,4]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4b() {\n  var htmlShow4b = document.getElementById(\"html-show4b\");\n  if (htmlShow4b.style.display === \"none\") {\n    htmlShow4b.style.display = \"block\";\n  } else {\n    htmlShow4b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4b\" style=\"display: none;\">\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i11)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">15<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">22<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">32<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">cprima<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">j<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o6} \\left[ \\mathop{-}3\\mathop{,}1\\mathop{,}1\\right] \\]\n<\/p>\n<p>\n\\[\\]\\[\\tag{%o11)\/R} \\left[ 22 l\\mathop{-}45 k\\mathop{-}96\\mathop{,}k\\mathop{,}\\mathop{-}\\left( 7 l\\right) \\mathop{+}15 k\\mathop{+}32\\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i12)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">y1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o12)\/R} \\left[ 22 l\\mathop{-}141\\mathop{,}1\\mathop{,}\\mathop{-}\\left( 7 l\\right) \\mathop{+}47\\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i13)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o13)\/R} \\left[ \\left[ \\mathop{-}53\\mathop{,}1\\mathop{,}19\\right] \\mathop{,}\\left[ \\mathop{-}31\\mathop{,}1\\mathop{,}12\\right] \\mathop{,}\\left[ \\mathop{-}9\\mathop{,}1\\mathop{,}5\\right] \\mathop{,}\\left[ 13\\mathop{,}1\\mathop{,}\\mathop{-}2\\right] \\right] \\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nObservemos que si consider\u00e1semos como par\u00e1metro la variable z, tendr\u00edamos la ecuaci\u00f3n 5x-15y=32-22z, y mcd(5,15)=5 no nos garantiza que 5|(32-22z) para cualquier z. Sin embargo, si ocurre en el caso de 22z=32(mod 5); es decir, 2z=2(mod 5), z=1+5k\n<\/div>\n<div class=\"comment\">\nEl resultado anterior no implican que no existan soluciones con \\(z\\neq 1+5k\\), de hecho hemos visto que [-9,1,5] o [13,1,-2] son soluciones. Sin embargo, no podemos llegar a dichas soluciones siguiendo el procedimiento de considerar la variable \\(z\\) como par\u00e1metro.\n<\/div>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nPor las indicaciones del ejercicios, supongamos que \\(z\\)=6&lt;10, entonces se plantea la ecuaci\u00f3n 5x-15y=-100\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i14)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">sol_z<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">diof2<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">15<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">100<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o14} \\left[ \\mathop{-}\\left( 3 k\\right) \\mathop{-}20\\mathop{,}\\mathop{-}k\\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i15)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">sol_z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o15} \\left[ \\left[ \\mathop{-}11\\mathop{,}3\\mathop{,}6\\right] \\mathop{,}\\left[ \\mathop{-}14\\mathop{,}2\\mathop{,}6\\right] \\mathop{,}\\left[ \\mathop{-}17\\mathop{,}1\\mathop{,}6\\right] \\mathop{,}\\left[ \\mathop{-}20\\mathop{,}0\\mathop{,}6\\right] \\mathop{,}\\left[ \\mathop{-}23\\mathop{,}\\mathop{-}1\\mathop{,}6\\right] \\mathop{,}\\left[ \\mathop{-}26\\mathop{,}\\mathop{-}2\\mathop{,}6\\right] \\right] \\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nAs\u00ed, la soluci\u00f3n que buscamos es [-17,1,6]\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i16)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">17<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o16} 61\\]\n<\/p>\n<\/div>\n<hr>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Resolver el sistema 3x+5y-z=12, 2x-3y+4z=3<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1dw() {\n  var htmlShow1dw = document.getElementById(\"html-show1dw\");\n  if (htmlShow1dw.style.display === \"none\") {\n    htmlShow1dw.style.display = \"block\";\n  } else {\n    htmlShow1dw.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1dw()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1dw\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<p>Planteemos las ecuaciones y simplifiquemos para dejar solo dos variables<\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i4)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">equ<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">12<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">eq1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">equ<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">equ<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o4)\/R} 17 y\\mathop{+}14 x\\mathop{=}51\\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i13)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">14<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">17<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">51<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">x1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">&quot;x=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">&quot;y=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o9} \\left[ \\mathop{-}6\\mathop{,}5\\mathop{,}1\\right] \\]\\[\\mbox{}\\\\\u00bbx=\u00bb17 k\\mathop{-}306\\mathop{<br \/>\n}\\]\\[\\mbox{}\\\\\u00bby=\u00bb255\\mathop{-}14 k\\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i16)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">s1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">equ<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">x1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o14} \\left[ z\\mathop{=}345\\mathop{-}19 k\\right] \\]\n<\/p>\n<p>\n\\[\\]\\[\\tag{%o16} \\left[ 17 k\\mathop{-}306\\mathop{,}255\\mathop{-}14 k\\mathop{,}345\\mathop{-}19 k\\right] \\]\n<\/p>\n<p>Ya tenemos la ecuaciones param\u00e9tricas como soluci\u00f3n del problema.<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Resolver el sistema 7x+2y+3z=5,5x+3y-2z=3 <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1c() {\n  var htmlShow1c = document.getElementById(\"html-show1c\");\n  if (htmlShow1c.style.display === \"none\") {\n    htmlShow1c.style.display = \"block\";\n  } else {\n    htmlShow1c.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1c\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i4)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">equ<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">eq1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">equ<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">equ<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o4)\/R} \\mathop{-}\\left( 29 z\\right) \\mathop{+}11 y\\mathop{=}\\mathop{-}4\\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i13)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">11<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">29<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">eucl_ext<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">z1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">c<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">&quot;y=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">&quot;z=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o9} \\left[ 8\\mathop{,}3\\mathop{,}1\\right] \\]\\[\\mbox{}\\\\\u00bby=\u00bb\\mathop{-}\\left( 29 k\\right) \\mathop{-}32\\mathop{<br \/>\n}\\]\\[\\mbox{}\\\\\u00bbz=\u00bb\\mathop{-}\\left( 11 k\\right) \\mathop{-}12\\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i16)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">s1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">equ<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">z1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">y1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">z1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o14} \\left[ x\\mathop{=}13 k\\mathop{+}15\\right] \\]\n<\/p>\n<p>\n\\[\\]\\[\\tag{%o16} \\left[ 13 k\\mathop{+}15\\mathop{,}\\mathop{-}\\left( 29 k\\right) \\mathop{-}32\\mathop{,}\\mathop{-}\\left( 11 k\\right) \\mathop{-}12\\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i18)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">k<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o17} \\left[ \\left[ k\\mathop{=}\\mathop{-}\\left( \\frac{15}{13}\\right) \\right] \\mathop{,}\\left[ k\\mathop{=}\\mathop{-}\\left( \\frac{32}{29}\\right) \\right] \\mathop{,}\\left[ k\\mathop{=}\\mathop{-}\\left( \\frac{12}{11}\\right) \\right] \\right] \\]\n<\/p>\n<p>\n\\[\\]\\[\\tag{%o18} \\left[ \\mathop{-}2\\mathop{,}\\mathop{-}2\\mathop{,}\\mathop{-}2\\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i19)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">sol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o19} \\left[ \\left[ \\mathop{-}24\\mathop{,}55\\mathop{,}21\\right] \\mathop{,}\\left[ \\mathop{-}11\\mathop{,}26\\mathop{,}10\\right] \\mathop{,}\\left[ 2\\mathop{,}\\mathop{-}3\\mathop{,}\\mathop{-}1\\right] \\right] \\]\n<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i20)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o20} \\left[ 5\\mathop{,}5\\mathop{,}5\\right] \\]\n<\/p>\n<\/div>\n<hr \/>\n","protected":false},"excerpt":{"rendered":"<p>Ecuaci\u00f3n de congruencias Recordemos, para resolver la ecuaci\u00f3n \\(aX\\equiv b {\\pmod {n}}\\), \\((aX \\equiv b (n))\\), cuando \\(\\mathbf{mcd}(a,n)=1\\), utilizamos bien soluci\u00f3n de B\u00e9zout, bien la funci\u00f3n \\(\\varphi\\) de Euler. Ejemplo: Resolver la&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[9],"class_list":["post-888","post","type-post","status-publish","format-standard","hentry","category-matematica-discreta","tag-practicas-mad"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/888","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=888"}],"version-history":[{"count":10,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/888\/revisions"}],"predecessor-version":[{"id":911,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/888\/revisions\/911"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=888"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=888"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=888"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}