{"id":773,"date":"2026-02-18T11:10:44","date_gmt":"2026-02-18T10:10:44","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=773"},"modified":"2026-02-22T19:29:05","modified_gmt":"2026-02-22T18:29:05","slug":"mad-el-mcd-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=773","title":{"rendered":"MAD: El mcd con maxima"},"content":{"rendered":"<p>El pasado d\u00eda vimos que el algoritmo de Euclides se fundamenta en el teorema:<\/p>\n<blockquote>\n<p><strong>Teorema<\/strong>: Si \\(a\\) y \\(b\\) son n\u00fameros enteros, \\[\\mathbf{mcd}(a,b)=\\mathbf{mcd}(b,r),\\] donde \\(r\\) es el resto del algoritmo de la divisi\u00f3n para \\(a\\) y \\(b\\) (\\(a=qb+r\\)).<\/p>\n<\/blockquote>\n<h3>Algoritmo para el c\u00e1lculo del \\(\\mathbf{mcd}\\)<\/h3>\n<p>\\[\\begin{array}{l} r=[b,\\mathbf{mod}(b,a)];\\\\ i=2;\\\\  \\mathbf{while}(r[i]&gt;0) \\\\ \\qquad r[i++]=\\mathbf{mod}(r[i-1],r[i]); \\\\ \\mathbf{end}\\\\ \\mathbf{print}(\u00ab\\mathbf{mcd}:\u00bb, r[i-1])\\end{array} \\]<\/p>\n<blockquote>\n<p><strong>Ejercicio: <\/strong> Crear una algoritmo que calcule el \\(\\mathbf{mcd}(a,b)\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv23a() {\n  var htmlShow23a = document.getElementById(\"html-show23a\");\n  if (htmlShow23a.style.display === \"none\") {\n    htmlShow23a.style.display = \"block\";\n  } else {\n    htmlShow23a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv23a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show23a\" 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\">(%i2)\t<\/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\">r1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">r1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">r2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">r3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">while<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r3<\/span><span class=\"code_endofline\">&gt;<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/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\">r1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">r2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">r3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">r3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">r2<\/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_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">21<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">15<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[3\\]<\/p>\n<\/div>\n<hr \/>\n<p>Este algoritmo nos calcula el mcd; sin embargo, para posteriores ejercicios utilizaremos el proceso pasado que vimos. Por ejemplo, para calcular el \\(\\mathbf{mcd}(43134,343)\\) construimos la siguiente tabla:<\/p>\n<table border=\"0\" cellspacing=\"0\" cellpadding=\"0\" class=\"ta1\">\n<colgroup>\n<col width=\"99\"\/>\n<col width=\"99\"\/>\n<col width=\"99\"\/>\n<col width=\"99\"\/>\n<col width=\"99\"\/><\/colgroup>\n<tr class=\"ro1\">\n<td style=\"text-align:left;width:2.258cm; \" class=\"Default\">\n<p>cociente<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>125<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>1<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>3<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>12<\/p>\n<\/td>\n<\/tr>\n<tr class=\"ro1\">\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>43134<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>343<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>259<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>84<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p style=\"color:#FF0000\";><strong>7<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr class=\"ro1\">\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>259<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>84<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>7<\/p>\n<\/td>\n<td style=\"text-align:right; width:2.258cm; \" class=\"Default\">\n<p>0<\/p>\n<\/td>\n<td style=\"text-align:left;width:2.258cm; \" class=\"Default\">\n<p>resto<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<p>Tanto la fila cociente como la fila resto las utilizaremos m\u00e1s adelante. Veamos c\u00f3mo obtenemos ambas filas con un algoritmo:<\/p>\n<h3>Algoritmo de Euclides<\/h3>\n<p>\\[\\begin{array}{l}<br \/>\nq=\\lfloor b\/a\\rfloor;\\\\<br \/>\nr=\\mathbf{mod}(b,a);\\\\<br \/>\ni=1;\\\\<br \/>\nn=a;\\\\<br \/>\n\\mathbf{while}(r[i]&gt;0) \\\\<br \/>\n\\qquad q[i++]=\\lfloor n\/r[i]\\rfloor ;\\\\<br \/>\n\\qquad r[i++]=\\mathbf{mod}(n,r[i]); \\\\<br \/>\n\\qquad n=r[i];\\\\<br \/>\n\\qquad i=i+1;\\\\<br \/>\n\\mathbf{end}\\\\<br \/>\n\\mathbf{print}(\u00abCocientes:\u00bb, q)\\\\<br \/>\n\\mathbf{print}(\u00abRestos:\u00bb, r)\\end{array} \\]<\/p>\n<blockquote>\n<p><strong>Ejercicio: <\/strong> Crear un algoritmo que calcule el \\(\\mathbf{mcd}(43134,343)\\) guardando los cocientes y los restos sucesivos.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv23b() {\n  var htmlShow23b = document.getElementById(\"html-show23b\");\n  if (htmlShow23b.style.display === \"none\") {\n    htmlShow23b.style.display = \"block\";\n  } else {\n    htmlShow23b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv23b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show23b\" 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\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">algEucl<\/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\">q<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r<\/span><span class=\"code_endofline\">,<\/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_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\/<\/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_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\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_function\">while<\/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_operator\">]<\/span><span class=\"code_endofline\">&gt;<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\">do<\/span><span class=\"code_operator\">(<\/span> \u00a0\u00a0 <span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/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\">n<\/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\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">n<\/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_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span> \u00a0\u00a0 <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_operator\">[<\/span><span class=\"code_variable\">q<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">]<\/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_function\">algEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">43134<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">343<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\left[ 125,1,3,12\\right] ,\\left[ 259,84,7,0\\right] \\right] \\]<\/p>\n<\/div>\n<hr \/>\n<h4>Procedimiento recursivo<\/h4>\n<blockquote>\n<p><strong>Ejercicio: <\/strong> Crear un algoritmo recursivo que calcule el \\(\\mathbf{mcd}\\).<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv23br() {\n  var htmlShow23br = document.getElementById(\"html-show23br\");\n  if (htmlShow23br.style.display === \"none\") {\n    htmlShow23br.style.display = \"block\";\n  } else {\n    htmlShow23br.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv23br()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show23br\" 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\">(%i2)<\/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\">if <\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> then <\/span><span class=\"code_variable\">a<\/span><span class=\"code_function\"> else <\/span><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/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><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">114<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">32<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\tag{%o2} 2\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio: <\/strong> Crear un algoritmo recursivo que calcule los cocientes en el c\u00e1lculo de \\(\\mathbf{mcd}\\).<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3e() {\n  var htmlShow3e = document.getElementById(\"html-show3e\");\n  if (htmlShow3e.style.display === \"none\") {\n    htmlShow3e.style.display = \"block\";\n  } else {\n    htmlShow3e.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3e()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3e\" 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(%i2)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">cEucl<\/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\">if<\/span> <span class=\"code_variable\">b<\/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_operator\">]<\/span> <span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">else<\/span> <span class=\"code_function\">cons<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_function\">cEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/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_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">cEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">43134<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">343<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\tag{%o2} \\left[ 125{,}1{,}3{,}12\\right] \\]\n<\/p>\n<\/div>\n<hr \/>\n<h3>mcm<\/h3>\n<p>Ya estamos en condiciones para determinar el \\(\\mathbf{mcm}\\) utilizando:<\/p>\n<blockquote>\n<p><strong>Teorema<\/strong>: Dados dos n\u00fameros enteros \\(a\\) y \\(b\\), entonces \\[\\mathbf{mcm}(a,b)=\\frac{|ab|}{\\mathbf{mcd}(a,b)}\\]<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejemplo: <\/strong> \u00bfCu\u00e1l es el valor de \\(\\mathbf{mcm}(30,25)\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2t() {\n  var htmlShow2t = document.getElementById(\"html-show2t\");\n  if (htmlShow2t.style.display === \"none\") {\n    htmlShow2t.style.display = \"block\";\n  } else {\n    htmlShow2t.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2t()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2t\" style=\"display: none;\">\n<!-- Text cell --> <\/p>\n<div class=\"comment\">Primero definimos el mcd:<\/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\">(%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\">if <\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> then <\/span><span class=\"code_variable\">a<\/span><span class=\"code_function\"> else <\/span><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/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\">$<\/span> <\/span>  <\/td>\n<\/tr>\n<\/table>\n<p> <!-- Text cell --> <\/p>\n<div class=\"comment\">Ahora aplicamos \\[\\mathbf{mcm}(a,b)=\\frac{|ab|}{\\mathbf{mcd}(a,b)}\\]<\/div>\n<p> <!-- Code cell --> <\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"> <span class=\"prompt\">(%i3)<\/span>  <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">mcm<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">abs<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_function\">mcm<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">30<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">25<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span> <\/span>  <\/td>\n<\/tr>\n<\/table>\n<p>\\[150\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Teorema de B\u00e9zout<\/h2>\n<h3>Con un proceso recursivo <\/h3>\n<blockquote>\n<p><strong>Ejemplo: <\/strong> Utilizar los cocientes y restos obtenidos del ejercicio anterior para resolver  \\[43134x+343y=\\mathbf{mcd}(43134,343)\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2y() {\n  var htmlShow2y = document.getElementById(\"html-show2y\");\n  if (htmlShow2y.style.display === \"none\") {\n    htmlShow2y.style.display = \"block\";\n  } else {\n    htmlShow2y.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2y()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2y\" style=\"display: none;\">\n<div class=\"comment\">\nTengamos en cuenta que \\(\\mathbf{mcd}(43134,343)=7\\). Consideremos a y b como constantes:\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(%i5)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">125<\/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_endofline\">,<\/span><span class=\"code_number\">12<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">r1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">r1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">r1<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\tag{%o5)\/R} 7\\mathop{=}\\mathop{-}\\left( 503 b\\right) \\mathop{+}4 a\\]\n<\/p>\n<p>Recordemos que \\(a=43134\\) y \\(b=343\\).\n<\/p><\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo: <\/strong> Sea v=[a,b], la  <strong>soluci\u00f3n de Bezout<\/strong> de \\(\\mathbf{mcd}(54180,47355)\\). \u00bfCu\u00e1nto es, en valor absoluto, el producto escalar [3,2].v\/||v||? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2rt() {\n  var htmlShow2rt = document.getElementById(\"html-show2rt\");\n  if (htmlShow2rt.style.display === \"none\") {\n    htmlShow2rt.style.display = \"block\";\n  } else {\n    htmlShow2rt.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2rt()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2rt\" 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(%i2)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<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\">if<\/span> <span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span> <span class=\"code_function\">then<\/span> <span class=\"code_variable\">a<\/span> <span class=\"code_function\">else<\/span> <span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/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\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">cEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span> <span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span> <span class=\"code_function\">if<\/span> <span class=\"code_variable\">b<\/span> <span class=\"code_operator\">=<\/span> <span class=\"code_number\">0<\/span> <span class=\"code_function\">then<\/span> <span class=\"code_operator\">[<\/span><span class=\"code_operator\">]<\/span> <span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">else<\/span> <span class=\"code_function\">cons<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_function\">cEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/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;\">\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\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">54180<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">47355<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">mcd<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/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 \/>\n<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">cEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\tag{d} 105\\]\n<\/p>\n<p>\n\\[\\tag{q} \\left[ 1\\mathop{,}6\\mathop{,}1\\mathop{,}15\\mathop{,}4\\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(%i11)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">r1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">a<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">r1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">r3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">r1<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">d<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">r2<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">\u00b7<\/span><span class=\"code_variable\">r3<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\tag{%o11)\/R} 105\\mathop{=}127 b\\mathop{-}111 a\\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nYa tenemos la soluci\u00f3n de Bezout que buscamos.\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\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_endofline\">\u2212<\/span><span class=\"code_number\">111<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">127<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_operator\">[<\/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 class=\"code_variable\">v<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">abs<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\tag{%o13} \\mathop{-}\\left( \\frac{79}{5 \\sqrt{1138}}\\right) \\]\n<\/p>\n<p>\n\\[\\tag{%o14} 0.4683666417157143\\]\n<\/p>\n<\/div>\n<hr \/>\n<h3>Con matrices<\/h3>\n<blockquote>\n<p><strong>Ejemplo: <\/strong> Sea v=[a,b], la  <strong>soluci\u00f3n de Bezout<\/strong> de \\(\\mathbf{mcd}(462,216)\\). \u00bfCu\u00e1nto es, en valor absoluto, el producto escalar [-1,5].v\/||v||? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2at() {\n  var htmlShow2at = document.getElementById(\"html-show2at\");\n  if (htmlShow2at.style.display === \"none\") {\n    htmlShow2at.style.display = \"block\";\n  } else {\n    htmlShow2at.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2at()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2at\" 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(%i2)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_function\">cEucl<\/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\">if<\/span> <span class=\"code_variable\">b<\/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_operator\">]<\/span> <span class=\"code_endofline\"><br \/>\n<\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_function\">else<\/span> <span class=\"code_function\">cons<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span> <span class=\"code_function\">cEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">b<\/span><span class=\"code_endofline\">,<\/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_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">cEucl<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">462<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">216<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\tag{q} \\left[ 2\\mathop{,}7\\mathop{,}5\\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(%i5)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">m<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">m<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/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{m} \\begin{pmatrix}\\mathop{-}7 &amp; 36\\\\<br \/>\n15 &amp; \\mathop{-}77\\end{pmatrix}\\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nYa tenemos la soluci\u00f3n de Bezout que buscamos.\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(%i8)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/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_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">abs<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\tag{%o7} \\frac{82}{\\sqrt{274}}\\]\n<\/p>\n<p>\n\\[\\tag{%o8} 4.953801165307452\\]\n<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo: <\/strong> Sea v=[a,b], la  <strong>soluci\u00f3n de Bezout<\/strong> de \\(\\mathbf{mcd}(43134,343)\\). \u00bfCu\u00e1nto es, en valor absoluto, el producto escalar [4,-3].v\/||v||? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2a() {\n  var htmlShow2a = document.getElementById(\"html-show2a\");\n  if (htmlShow2a.style.display === \"none\") {\n    htmlShow2a.style.display = \"block\";\n  } else {\n    htmlShow2a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a\" style=\"display: none;\">\nPor el ejercicio anterior sabemos que<br \/>\n<!-- 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\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">125<\/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_endofline\">,<\/span><span class=\"code_number\">12<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/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\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">l<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span>(<span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">0<\/span>,<span class=\"code_number\">1<\/span>],[<span class=\"code_number\">1<\/span>,<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">q<\/span>[<span class=\"code_number\">1<\/span>][<span class=\"code_variable\">i<\/span>]]),<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_function\">length<\/span>(<span class=\"code_variable\">q<\/span>[<span class=\"code_number\">1<\/span>]))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_variable\">l2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">0<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">l3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span>(<span class=\"code_variable\">l2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">l2<\/span>.<span class=\"code_variable\">l<\/span>[<span class=\"code_variable\">i<\/span>],<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_function\">length<\/span>(<span class=\"code_variable\">l<\/span>))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abSoluci\u00f3n de Bezout: (\u00ab<\/span>,<span class=\"code_variable\">l3<\/span>[<span class=\"code_number\">4<\/span>][<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>],<span class=\"code_string\">\u00ab,\u00bb<\/span>,<span class=\"code_variable\">l3<\/span>[<span class=\"code_number\">4<\/span>][<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>],<span class=\"code_string\">\u00ab)\u00bb<\/span>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(l)<\/mtext><\/mtd><mtd><mo>[<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>125<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>3<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>12<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(l3)<\/mtext><\/mtd><mtd><mo>[<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>125<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mrow><mi>\u2212<\/mi><mn>125<\/mn><\/mrow><\/mtd><mtd><mn>126<\/mn><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><mtd><mn>4<\/mn><\/mtd><\/mtr><mtr><mtd><mn>126<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>503<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>,<\/mo><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mn>4<\/mn><\/mtd><mtd><mrow><mi>\u2212<\/mi><mn>49<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mrow><mi>\u2212<\/mi><mn>503<\/mn><\/mrow><\/mtd><mtd><mn>6162<\/mn><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><mo>]<\/mo><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"center\"><mtd><mtext\/><\/mtd><mtd><mo>Soluci\u00f3n de Bezout: (<\/mo><mo\/><mn>4<\/mn><mo\/><mo>,<\/mo><mo\/><mi>\u2212<\/mi><mn>503<\/mn><mo\/><mo>)<\/mo><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">l3<\/span>[<span class=\"code_number\">4<\/span>][<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">l3<\/span>[<span class=\"code_number\">4<\/span>][<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>]]<span class=\"code_endofline\">$<\/span><br \/>[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>].<span class=\"code_variable\">v<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">v<\/span>.<span class=\"code_variable\">v<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">abs<\/span>(<span class=\"code_variable\">%<\/span>),<span class=\"code_variable\">numer<\/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=\"center\"><mtd><mtext>(%o8) <\/mtext><\/mtd><mtd><mfrac><mn>305<\/mn><msqrt><mn>10121<\/mn><\/msqrt><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(%o9) <\/mtext><\/mtd><mtd><mn>3.03171328560307<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<h3>Algoritmo extendido de Euclides<\/h3>\n<p>El proceso que hemos calculado con matrices lo podemos programar de la siguiente forma:<br \/>\n<!-- Code cell --><\/p>\n<table>\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\">eucl_ext<\/span>(<span class=\"code_variable\">a<\/span>,<span class=\"code_variable\">b<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">block<\/span>([<span class=\"code_variable\">s<\/span>,<span class=\"code_variable\">t<\/span>,<span class=\"code_variable\">r<\/span>],<br \/> \u00a0\u00a0 <span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">0<\/span>],<br \/> \u00a0\u00a0 <span class=\"code_variable\">t<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">0<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>],<br \/> \u00a0\u00a0 <span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">a<\/span>,<span class=\"code_variable\">b<\/span>],<br \/> \u00a0\u00a0 while (<span class=\"code_variable\">r<\/span>[<span class=\"code_number\">2<\/span>]&gt;<span class=\"code_number\">0<\/span>) do ( \u00a0\u00a0\u00a0<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">s<\/span>[<span class=\"code_number\">3<\/span>]<span class=\"code_operator\">:<\/span><span class=\"code_variable\">s<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">&#8211;<\/span><span class=\"code_function\">floor<\/span>(<span class=\"code_variable\">r<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">\/<\/span><span class=\"code_variable\">r<\/span>[<span class=\"code_number\">2<\/span>])<span class=\"code_operator\">*<\/span><span class=\"code_variable\">s<\/span>[<span class=\"code_number\">2<\/span>],<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">t<\/span>[<span class=\"code_number\">3<\/span>]<span class=\"code_operator\">:<\/span><span class=\"code_variable\">t<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">&#8211;<\/span><span class=\"code_function\">floor<\/span>(<span class=\"code_variable\">r<\/span>[<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">\/<\/span><span class=\"code_variable\">r<\/span>[<span class=\"code_number\">2<\/span>])<span class=\"code_operator\">*<\/span><span class=\"code_variable\">t<\/span>[<span class=\"code_number\">2<\/span>],<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">s<\/span>[<span class=\"code_number\">2<\/span>],<span class=\"code_variable\">s<\/span>[<span class=\"code_number\">3<\/span>],<span class=\"code_number\">0<\/span>],<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">t<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">t<\/span>[<span class=\"code_number\">2<\/span>],<span class=\"code_variable\">t<\/span>[<span class=\"code_number\">3<\/span>],<span class=\"code_number\">0<\/span>],<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">r<\/span>[<span class=\"code_number\">2<\/span>],<span class=\"code_function\">mod<\/span>(<span class=\"code_variable\">r<\/span>[<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">r<\/span>[<span class=\"code_number\">2<\/span>])]<br \/> \u00a0\u00a0 ),<br \/> \u00a0\u00a0 [<span class=\"code_variable\">s<\/span>[<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">t<\/span>[<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">r<\/span>[<span class=\"code_number\">1<\/span>]]<br \/>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<hr\/>\n<blockquote>\n<p><strong>Ejemplo: <\/strong> Sea v=[a,b], la  <strong>soluci\u00f3n de Bezout<\/strong> de \\(\\mathbf{mcd}(5432,872)\\). \u00bfCu\u00e1nto es, en valor absoluto, el producto escalar [3,-5].v\/||v||? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2at9() {\n  var htmlShow2at9 = document.getElementById(\"html-show2at9\");\n  if (htmlShow2at9.style.display === \"none\") {\n    htmlShow2at9.style.display = \"block\";\n  } else {\n    htmlShow2at9.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2at9()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2at9\" style=\"display: none;\">\n<p>Calculemos los restos y los cocientes que utilizaremos posteriormente.<\/p>\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\">(%i9) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">5432<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">872<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">floor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">m<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">r<\/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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">r<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">mod<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">m<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">r<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><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 <span class=\"code_variable\">q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">floor<\/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_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/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_operator\">]<\/span><span class=\"code_operator\">)<\/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_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_operator\">\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><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">q<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">r<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ 6, 4, 2, 1, 3, 2\\right] \\]<\/p>\n<p>\\[\\left[ 200, 72, 56, 16, 8, 0\\right] \\]<\/p>\n<p>Declaremos la funci\u00f3n que nos permita aplicar el algoritmo extendido de Euclides para encontrar la soluci\u00f3n que buscamos.<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">qm<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>Apliquemos el algoritmo extendido de Euclides multiplicando las matrices generadas con los cocientes.<\/p>\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_variable\">M<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ident<\/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_endofline\"><br \/><\/span><span class=\"code_variable\">ml<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">M<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">M<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">qm<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/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_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\begin{bmatrix}0 &amp; 1\\\\1 &amp; -6\\end{bmatrix}, \\begin{bmatrix}1 &amp; -4\\\\-6 &amp; 25\\end{bmatrix}, \\begin{bmatrix}-4 &amp; 9\\\\25 &amp; -56\\end{bmatrix}, \\begin{bmatrix}9 &amp; -13\\\\-56 &amp; 81\\end{bmatrix}, \\begin{bmatrix}-13 &amp; 48\\\\81 &amp; -299\\end{bmatrix}, \\begin{bmatrix}48 &amp; -109\\\\-299 &amp; 679\\end{bmatrix}\\right] \\]<\/p>\n<p>Tenemos la soluci\u00f3n, la primera columna de la \u00faltima matriz. Resolvamos el ejercicio.<\/p>\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\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">col<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">ml<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">q<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/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_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[5.412307287160986\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Funci\u00f3n \\(\\varphi\\) de Euler<\/h2>\n<blockquote><p>\n<strong>Ejercicio:<\/strong> Para \\(n\\in\\mathbb{Z}^+\\), se define la funci\u00f3n \\(\\varphi\\) de Euler como \\[\\varphi (n)=|\\{m\\in\\mathbb{Z}^+|m&lt;n, mcd(n,m)=1\\}|.\\]<br \/>\nCrear una funci\u00f3n en maxima que determine \\(\\varphi(n)\\) para \\(n\\in\\mathbb{Z}^+\\).\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2a45() {\n  var htmlShow2a45 = document.getElementById(\"html-show2a45\");\n  if (htmlShow2a45.style.display === \"none\") {\n    htmlShow2a45.style.display = \"block\";\n  } else {\n    htmlShow2a45.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2a45()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a45\" 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\">fi<\/span>(<span class=\"code_variable\">n<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">block<\/span>([<span class=\"code_variable\">f<\/span>,<span class=\"code_variable\">i<\/span>],<br \/> \u00a0\u00a0 <span class=\"code_variable\">f<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span>,<br \/> \u00a0\u00a0 for <span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span> thru <span class=\"code_variable\">n<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">1<\/span> do(<br \/> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 if (<span class=\"code_function\">mcd<\/span>(<span class=\"code_variable\">i<\/span>,<span class=\"code_variable\">n<\/span>)<span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span>) then <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><br \/> \u00a0\u00a0 ),<br \/> \u00a0\u00a0 <span class=\"code_variable\">f<\/span><br \/>)<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<hr \/>\n<h2>Ecuaciones diof\u00e1nticas de dos variables<\/h2>\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>&nbsp;<\/p>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong> \u00bfCu\u00e1l es la suma de d\u00edgitos del \\(\\textbf{mcd}(1108955617, 2283735839)\\)?<\/p>\n<div id=\"menu-a\">\n<ul>\n<li>21<\/li>\n<li>14<\/li>\n<li>6<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p id=\"htmlContent\" class=\"text-html\"><strong>B.)<\/strong><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>El pasado d\u00eda vimos que el algoritmo de Euclides se fundamenta en el teorema: Teorema: Si \\(a\\) y \\(b\\) son n\u00fameros enteros, \\[\\mathbf{mcd}(a,b)=\\mathbf{mcd}(b,r),\\] donde \\(r\\) es el resto del algoritmo de 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-773","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\/773","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=773"}],"version-history":[{"count":20,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/773\/revisions"}],"predecessor-version":[{"id":865,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/773\/revisions\/865"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=773"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=773"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}