{"id":262,"date":"2018-04-20T08:45:00","date_gmt":"2018-04-20T07:45:00","guid":{"rendered":"http:\/\/clases.jesussoto.es\/?p=262"},"modified":"2018-04-20T08:53:45","modified_gmt":"2018-04-20T07:53:45","slug":"mad-ecuacion-de-congruencias","status":"publish","type":"post","link":"https:\/\/curso17.jesussoto.es\/?p=262","title":{"rendered":"MAD: Ecuaci\u00f3n de congruencias"},"content":{"rendered":"<p>Los pasados d\u00edas hemos trabajado en los cimientos para abordar la ecuaci\u00f3n de congruencias $$aX\\equiv b (n)$$<\/p>\n<p>Ahora podemos establecer los criterios que nos permitir\u00e1n conocer cu\u00e1ndo existe soluci\u00f3n:<\/p>\n<blockquote>\n<p><em>La ecuaci\u00f3n $aX \\equiv b (n)$ tiene soluci\u00f3n si, y s\u00f3lo si, el $mcd(a,n)|b$<\/em><\/p>\n<\/blockquote>\n<p>El procedimiento m\u00e1s sencillo es cuando $mcd(a,n)=1$, que en cuyo caso siempre tiene soluci\u00f3n y esta se obtiene buscando el inverso de $a$ en $\\mathbb{Z}_n$.<\/p>\n<p>\u00bfQu\u00e9 ocurre si $mcd(a,n)=d$ y $d|b$? En tal caso la soluci\u00f3n que buscamos depender\u00e1 de la soluci\u00f3n de<\/p>\n<p>$$\\frac{a}{d}X_0 \\equiv \\frac{b}{d} \\left(\\frac{n}{d}\\right)$$<\/p>\n<p>En tal caso, las soluciones ser\u00e1 varias y vendr\u00e1s dadas por:<\/p>\n<p>$$X\\equiv \\left[X_0+\\frac{n}{d}k\\right](n), $$<\/p>\n<p>donde $k\\in \\{0,1,2,\\ldots,d-1\\}$.<\/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> Resolver 6X\u2261 11 (35)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Los pasados d\u00edas hemos trabajado en los cimientos para abordar la ecuaci\u00f3n de congruencias $$aX\\equiv b (n)$$ Ahora podemos establecer los criterios que nos permitir\u00e1n conocer cu\u00e1ndo existe soluci\u00f3n: La ecuaci\u00f3n $aX \\equiv b (n)$ tiene soluci\u00f3n si, y s\u00f3lo si, el $mcd(a,n)|b$ El procedimiento m\u00e1s sencillo es cuando $mcd(a,n)=1$, que en cuyo caso siempre&hellip; <a class=\"more-link\" href=\"https:\/\/curso17.jesussoto.es\/?p=262\">Seguir leyendo <span class=\"screen-reader-text\">MAD: Ecuaci\u00f3n de congruencias<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[5],"tags":[],"_links":{"self":[{"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/262"}],"collection":[{"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=262"}],"version-history":[{"count":1,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/262\/revisions"}],"predecessor-version":[{"id":263,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/262\/revisions\/263"}],"wp:attachment":[{"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=262"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=262"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}