{"id":954,"date":"2020-09-20T20:29:23","date_gmt":"2020-09-20T20:29:23","guid":{"rendered":"http:\/\/localhost\/?page_id=954"},"modified":"2022-05-30T08:12:33","modified_gmt":"2022-05-30T11:12:33","slug":"exercicios-8","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/algebra-a\/exercicios-8\/","title":{"rendered":"Exerc\u00edcios 8"},"content":{"rendered":"<p>1. O Teorema de <a href=\"https:\/\/en.wikipedia.org\/wiki\/John_Wilson_(mathematician)\">Wilson<\/a> afirma que um n\u00famero natural $n\\geq 2$ \u00e9 primo se e somente se $(n-1)!\\equiv -1\\pmod n$. Demonstre Wilson&#8217;s Theorem.<\/p>\n<p>2. Mostre, usando o Teorema de Fermat, que $2^{70}+3^{70}$ \u00e9 divis\u00edvel por 13.<\/p>\n<p>3. Mostre que a equa\u00e7\u00e3o $x^{13}+12x+13y^6=1$ n\u00e3o admite solu\u00e7\u00f5es inteiras. [Dica: reduze m\u00f3dulo 13 e use o Pequeno Teorema de Fermat.]<\/p>\n<p><!--4. Teorema de Euler: Se $n\\geq 2$ \u00e9 um n\u00famero natural e $k\\in\\mathbb Z$ tais que $\\mbox{mdc}(n,k)=1$,\u00a0 ent\u00e3o $k^{\\varphi(n)}\\equiv 1\\pmod n$. Demonstre o Teorema de Euler. [Dica: Assuma que $U(n)=\\{a_1,\\ldots,a_{\\varphi(n)}\\}$. Mostre para $a\\in U(n)$ que $\\{aa_1,\\ldots,aa_{\\varphi(n)}\\}= U(n)$ e conclua que $\\prod_{i} a_i=\\prod_i aa_i$.]\n--><\/p>\n<p>4. Seja $a$ um n\u00famero inteiro escrito na base $10$. Mostre que o \u00faltimo algarismo de $n$ \u00e9 igual ao \u00faltimo algarismo de $n^5$.<\/p>\n<p>5. Seja $p=4k+3$ um primo e considere a equa\u00e7\u00e3o $\\overline x^2= \\overline a$ sobre $\\mathbb Z_p$. Mostre que se a equa\u00e7\u00e3o possui solu\u00e7\u00e3o, ent\u00e3o $a^{k+1}$ e $-a^{k+1}$ ser\u00e3o solu\u00e7\u00f5es.<\/p>\n<p>6. (Ol\u00edmpiada de Matem\u00e1tica de Paraguay, 2009). Conte os n\u00fameros $n$ entre<br \/>\n$1$ e $2022$ tal que o \u00faltimo d\u00edgito de $n^{20}$ \u00e9 1.<br \/>\n[Dica: Use o Teorema de Euler para caraterizar os n\u00fameros $n$ para os quais o \u00faltimo d\u00edgito de $n^{20}$ \u00e9 1 veja tamb\u00e9m o <a href=\"http:\/\/youtu.be\/iDrHPrmYG-0\">v\u00eddeo de Michael Penn<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1. O Teorema de Wilson afirma que um n\u00famero natural $n\\geq 2$ \u00e9 primo se e somente se $(n-1)!\\equiv -1\\pmod n$. Demonstre Wilson&#8217;s Theorem. 2. Mostre, usando o Teorema de Fermat, que $2^{70}+3^{70}$ \u00e9 divis\u00edvel por 13. 3. Mostre que a equa\u00e7\u00e3o $x^{13}+12x+13y^6=1$ n\u00e3o admite solu\u00e7\u00f5es inteiras. [Dica: reduze m\u00f3dulo 13 e use o Pequeno &hellip; <a href=\"http:\/\/localhost\/index.php\/ensino\/algebra-a\/exercicios-8\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Exerc\u00edcios 8<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":706,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/954"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=954"}],"version-history":[{"count":5,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/954\/revisions"}],"predecessor-version":[{"id":956,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/954\/revisions\/956"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/706"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}