{"id":260,"date":"2018-04-19T08:30:59","date_gmt":"2018-04-19T07:30:59","guid":{"rendered":"http:\/\/clases.jesussoto.es\/?p=260"},"modified":"2018-04-18T08:20:34","modified_gmt":"2018-04-18T07:20:34","slug":"mad-funcion-%cf%86-de-euler","status":"publish","type":"post","link":"https:\/\/curso17.jesussoto.es\/?p=260","title":{"rendered":"MAD: Funci\u00f3n \u03c6 de Euler"},"content":{"rendered":"<p>En el d\u00eda de hoy hemos visto la funci\u00f3n $\\varphi $ de Euler. Esta funci\u00f3n se define como<\/p>\n<p>$$\\varphi (n)=|\\{m\\in\\mathbb{Z}^+|m&lt;n, mcd(n,m)=1\\}|.$$<\/p>\n<p>Esta funci\u00f3n cumple propiedades muy interesantes, como<\/p>\n<ul>\n<li>Si $p$ es primo, $\\varphi (p)=p-1$<\/li>\n<li>Si $p$ es primo, $\\varphi (p^\\alpha)=p^{\\alpha -1}(p-1)$<\/li>\n<li>Si $mcd(n,m)=1$ es $\\varphi (nm)=\\varphi (n)\\varphi (m)$<\/li>\n<\/ul>\n<p>Estos resultados nos sirven para exponer el Teorema de Euler:<\/p>\n<blockquote><p>Si $a,n\\in\\mathbb{Z}$ con $mcd(n,a)=1$, entonces $a^{\\varphi (n)}\\equiv 1(n)$<\/p><\/blockquote>\n<p>Este resultado ofrece adem\u00e1s una forma de obtener el inverso de un n\u00famero en $\\mathbb{Z}_n$, siempre que este exista.<\/p>\n<p>La descomposici\u00f3n de un numero entero en sus factores primos nos permite formular un resultado pr\u00e1ctico para calcular la funci\u00f3n $\\varphi$<\/p>\n<blockquote><p>Si $n\\in\\mathbb{Z}$ es ${\\displaystyle n=p_{1}^{k_{1}}\\cdots p_{r}^{k_{r}}}$, entonces $$\\varphi (n)=n\\sum_{i=1}^r\\left(1-\\frac{1}{p_i}\\right)$$<\/p><\/blockquote>\n<p>$\\quad $<\/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 $123321^{123}\\equiv X(36)$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>En el d\u00eda de hoy hemos visto la funci\u00f3n $\\varphi $ de Euler. Esta funci\u00f3n se define como $$\\varphi (n)=|\\{m\\in\\mathbb{Z}^+|m&lt;n, mcd(n,m)=1\\}|.$$ Esta funci\u00f3n cumple propiedades muy interesantes, como Si $p$ es primo, $\\varphi (p)=p-1$ Si $p$ es primo, $\\varphi (p^\\alpha)=p^{\\alpha -1}(p-1)$ Si $mcd(n,m)=1$ es $\\varphi (nm)=\\varphi (n)\\varphi (m)$ Estos resultados nos sirven para exponer el&hellip; <a class=\"more-link\" href=\"https:\/\/curso17.jesussoto.es\/?p=260\">Seguir leyendo <span class=\"screen-reader-text\">MAD: Funci\u00f3n \u03c6 de Euler<\/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\/260"}],"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=260"}],"version-history":[{"count":4,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/260\/revisions"}],"predecessor-version":[{"id":266,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/260\/revisions\/266"}],"wp:attachment":[{"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=260"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=260"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/curso17.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}