{"id":166957,"date":"2020-10-20T21:40:55","date_gmt":"2020-10-21T00:40:55","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=166957"},"modified":"2023-07-24T06:23:51","modified_gmt":"2023-07-24T09:23:51","slug":"problema-para-ajudar-na-escola-ultimo-digito","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-ultimo-digito\/","title":{"rendered":".Problema para ajudar na escola: \u00daltimo d\u00edgito"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:25px;padding-bottom:15px; margin-bottom:40px;\">\n<p class=\"tit\" style=\"line-height:20px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#00BF00;\"> (A partir do 9\u00ba ano do E. F.- <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: M\u00e9dio<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nDetermine o algarismo das unidades da soma [tex] \\boxed{3^{2020}+4^{2020}}\\,[\/tex].<\/p>\n<p style=\"font-size: 12px;text-align:right;line-height:10px;\">Adaptado do <strong> 6\u00ba Campeonato de Matem\u00e1tica de la Universidad de La Frontera<\/strong>, 2013.<\/p>\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"cb31 g1n\" style=\"margin:30px;line-height:25px; width:90%; margin-left:auto; margin-right:auto; \">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" style=\"border:none\"  alt=\"explicador_p\" width=\"63\" height=\"76\" class=\"alignleft size-full wp-image-1001\" \/><\/p>\n<p class=\"c3 ti3\" style=\"line-height:20px;margin:10px\">  Ajuda <\/p>\n<p style=\"text-align:center\"><strong>Algoritmo de Euclides<\/strong> ou <strong>Divis\u00e3o Euclidiana<\/strong><\/p>\n<p>Sejam [tex]a[\/tex] e [tex]b[\/tex] n\u00fameros naturais, com [tex]b\\ne 0.[\/tex] <\/p>\n<p style=\"margin:20px\">\n<table class=\"bf\" style=\" width:90%;\">\n<tr>\n<td class=\"bf\">\n[tex]\\qquad \\qquad \\begin{array}{r}<br \/>\na \\, \\end{array}   \\begin{array}{|r}<br \/>\n \\, b \\,  \\,  \\,  \\\\  \\hline<br \/>\n\\end{array}[\/tex]<br \/>\n[tex]\\qquad \\qquad\\begin{array}{r}<br \/>\nr<br \/>\n\\end{array}\\begin{array}{r}<br \/>\n  \\,  \\,  \\, q<br \/>\n\\end{array}\\qquad \\qquad[\/tex]<\/td>\n<td class=\"bf\">\nAo dividirmos  [tex]a[\/tex] por [tex]b[\/tex] encontraremos um quociente [tex]q\\, [\/tex]  e um resto [tex]\\,r [\/tex], naturais e \u00fanicos, tais que:<br \/>\n [tex] \\,  \\,  \\, \\\\ \\textcolor{#800000}{(1)} \\,  \\,  \\,  \\,  \\,  \\, 0 \\le r \\lt b  \\,  \\,  \\,  \\,  \\,  \\,  \\,\\qquad  \\textcolor{#800000}{(2)} \\,  \\, a=b\\cdot q+r.[\/tex]<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 1px; margin-bottom: 20px;border: 0 none\"\/>\nVamos analisar separadamente os algarismos das unidades das pot\u00eancias [tex] 3^{2020}\\,[\/tex] e [tex] \\,4^{2020}\\,[\/tex].<br \/>\nObserve que:<\/p>\n<ul>\n<li>[tex]3^1=3[\/tex]; assim, o algarismo das unidades de [tex]3^\\textcolor{red}{1}[\/tex] \u00e9 [tex]\\textcolor{red}{3}[\/tex].<\/li>\n<li>[tex]3^2=9[\/tex]; assim, o algarismo das unidades de [tex]3^\\textcolor{#00BF00}{2}[\/tex] \u00e9 [tex]\\textcolor{#00BF00}{9}[\/tex].<\/li>\n<li>[tex]3^3=27[\/tex]; assim, o algarismo das unidades de [tex]3^\\textcolor{#FF00FF}{3}[\/tex] \u00e9 [tex]\\textcolor{#FF00FF}{7}[\/tex].<\/li>\n<li>[tex]3^4=81[\/tex]; assim, o algarismo das unidades de [tex]3^\\textcolor{#3BB9FF}{4}[\/tex] \u00e9 [tex]\\textcolor{#3BB9FF}{1}[\/tex].<\/li>\n<li>[tex]3^5=243[\/tex]; assim, o algarismo das unidades de [tex]3^\\textcolor{red}{5}[\/tex] \u00e9 [tex]\\textcolor{red}{3}[\/tex].<\/li>\n<li>[tex]3^6=729[\/tex]; assim, o algarismo das unidades de [tex]3^\\textcolor{#00BF00}{6}[\/tex] \u00e9 [tex]\\textcolor{#00BF00}{9}[\/tex].<\/li>\n<\/ul>\n<p>Note que os algarismos das unidades das pot\u00eancias de [tex]3[\/tex] s\u00e3o &quot;[tex]3,\\; 9,\\, 7\\,[\/tex] ou [tex]1[\/tex]&quot; e se repetem ciclicamente, nessa ordem, de quatro em quatro. Assim, conseguimos estabelecer o seguinte padr\u00e3o:<\/p>\n<p style=\"text-align:center;font-size:13px\">[tex]\\begin{array}{c|c| l}<br \/>\n\\text{algarismo das unidades}&#038; \\text{expoente}&#038; \\text{forma geral do expoente}\\\\<br \/>\n\\hline<br \/>\n\\textcolor{red}{3}&#038; n=\\textcolor{red}{1},\\,1+4=\\textcolor{red}{5},\\,5+4=\\textcolor{red}{9},\\cdots&#038;\\quad  \\textcolor{red}{n=4k+1},\\,k\\in \\mathbb{N}\\\\<br \/>\n\\hline<br \/>\n\\textcolor{#00BF00}{9}&#038; n=\\textcolor{#00BF00}{2},\\,2+4=\\textcolor{#00BF00}{6},\\,6+4=\\textcolor{#00BF00}{10},\\cdots&#038;\\quad  \\textcolor{#00BF00}{n=4k+2},\\,k\\in \\mathbb{N}\\\\<br \/>\n\\hline<br \/>\n\\textcolor{#FF00FF}{7}&#038; n=\\textcolor{#FF00FF}{3},\\,3+4=\\textcolor{#FF00FF}{7},\\,7+4=\\textcolor{#FF00FF}{11},\\cdots&#038;\\quad  \\textcolor{#FF00FF}{n=4k+3},\\,k\\in \\mathbb{N}\\\\<br \/>\n\\hline<br \/>\n\\textcolor{#3BB9FF}{1}&#038; n=\\textcolor{#3BB9FF}{4},\\,4+4=\\textcolor{#3BB9FF}{8},\\,8+4=\\textcolor{#3BB9FF}{12},\\cdots&#038;\\quad \\textcolor{#3BB9FF}{n=4k},\\,k\\in \\mathbb{N}^*\\\\<br \/>\n\\hline<br \/>\n\\end{array}<br \/>\n\\;[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p>Agora, com rela\u00e7\u00e3o \u00e0s pot\u00eancias de [tex]4[\/tex], observe que:<\/p>\n<ul>\n<li>[tex]4^1=4[\/tex]; assim, o algarismo das unidades de [tex]4^\\textcolor{#A23BEC}{1}[\/tex] \u00e9 [tex]\\textcolor{#A23BEC}{4}[\/tex].<\/li>\n<li>[tex]4^2=16[\/tex]; assim, o algarismo das unidades de [tex]4^\\textcolor{#F87217}{2}[\/tex] \u00e9 [tex]\\textcolor{#F87217}{6}[\/tex].<\/li>\n<li>[tex]4^3=64[\/tex]; assim, o algarismo das unidades de [tex]4^\\textcolor{#A23BEC}{3}[\/tex] \u00e9 [tex]\\textcolor{#A23BEC}{4}[\/tex].<\/li>\n<li>[tex]4^4=256[\/tex]; assim, o algarismo das unidades de [tex]4^\\textcolor{#F87217}{4}[\/tex] \u00e9 [tex]\\textcolor{#F87217}{6}[\/tex].<\/li>\n<\/ul>\n<p>Note que os algarismos das unidades das pot\u00eancias de [tex]4[\/tex] s\u00e3o &quot;[tex]4\\,[\/tex] ou [tex]6[\/tex]&quot; e se repetem ciclicamente de dois em dois; com isso, conseguimos estabelecer o seguinte padr\u00e3o:<\/p>\n<p style=\"text-align:center;font-size:13px\">[tex]\\begin{array}{c|c| l}<br \/>\n\\text{algarismo das unidades}&#038; \\text{expoente}&#038; \\text{forma geral do expoente}\\\\<br \/>\n\\hline<br \/>\n\\textcolor{#A23BEC}{4}&#038; n=\\textcolor{#A23BEC}{1},\\,1+2=\\textcolor{#A23BEC}{3},\\,3+2=\\textcolor{#A23BEC}{5},\\cdots&#038;\\quad  \\textcolor{#A23BEC}{n \\text{ \u00edmpar}}\\\\<br \/>\n\\hline<br \/>\n\\textcolor{#F87217}{6}&#038; n=\\textcolor{#F87217}{2},\\,2+2=\\textcolor{#F87217}{4},\\,4+2=\\textcolor{#F87217}{6},\\cdots&#038;\\quad \\textcolor{#F87217}{ n\\text{ par}}\\\\<br \/>\n\\hline<br \/>\n\\end{array}<br \/>\n\\;[\/tex]<\/p>\n<p style=\"margin:20px\">\n<table class=\"bf\">\n<tr>\n<td class=\"bf\">\n[tex]\\qquad \\qquad \\begin{array}{r}<br \/>\n2020 \\, \\end{array}   \\begin{array}{|r}<br \/>\n \\,\\,4 \\,  \\,  \\, \\,  \\, \\\\  \\hline<br \/>\n\\end{array}[\/tex]<br \/>\n[tex]\\qquad \\qquad\\begin{array}{r}<br \/>\n\\quad \\,  \\, 0<br \/>\n\\end{array}\\begin{array}{r}<br \/>\n  \\,  \\,  \\, 505<br \/>\n\\end{array}\\qquad \\qquad[\/tex]<\/td>\n<td class=\"bf\">\nA divis\u00e3o de [tex]2020[\/tex] por [tex]4[\/tex], nos indica que [tex]2020[\/tex] \u00e9 m\u00faltiplo de [tex]4[\/tex], ou seja, da forma [tex]\\textcolor{#3BB9FF}{2020=4 \\cdot 505}\\,.[\/tex]<br \/>\nAssim, o algarismo das unidades da pot\u00eancia  [tex]3^\\textcolor{#3BB9FF}{2020}[\/tex] \u00e9 [tex]\\textcolor{#3BB9FF}{1}\\,.[\/tex] <\/td>\n<\/tr>\n<\/table>\n<p>Por outro lado, [tex]2020[\/tex] \u00e9 um n\u00famero par; logo, o algarismo das unidades da pot\u00eancia [tex]4^\\textcolor{#F87217}{2020}[\/tex] \u00e9 [tex]\\textcolor{#F87217}{6}\\,.[\/tex]<br \/>\nDessa forma, podemos esquematizar a soma [tex] \\boxed{3^{2020}+4^{2020}}\\,[\/tex] da seguinte forma:<br \/>\n[tex]\\qquad \\qquad \\begin{array}{r l}<br \/>\n&#038;\\cdots 1\\\\<br \/>\n+&#038;\\cdots 6\\\\<br \/>\n\\hline<br \/>\n&#038;\\cdots 7<br \/>\n\\end{array}[\/tex]<br \/>\ne, portanto, o algarismo das unidades da soma [tex] \\boxed{3^{2020}+4^{2020}}\\,[\/tex] \u00e9 [tex]\\,\\fcolorbox{black}{#eee0e5}{$7$}\\,.[\/tex]<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:20px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto;width:640px;\">\n<tr>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"80\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:520px;background-color: #FFF5EE; padding:10px; font-family: calibri; font-size: 13px;color: #1C1C1C; line-height: 20px; \"> Se for conveniente, voc\u00ea pode obter um arquivo PDF desta p\u00e1gina, com o problema e a solu\u00e7\u00e3o, clicando no bot\u00e3o abaixo.<br \/>\nVoc\u00ea pode abrir o arquivo diretamente no seu navegador (<em>Chrome<\/em>, <em>Edge<\/em>, <em>Firefox<\/em>, <em>Safari<\/em>, entre outros), mas tamb\u00e9m pode utilizar o software gratuito <em><strong>Adobe Acrobat Reader<\/strong><\/em>.<br \/>\nCaso o dispositivo que voc\u00ea est\u00e1 utilizando n\u00e3o tenha o <strong><em>Acrobat Reader<\/em><\/strong> instalado, \u00e9 s\u00f3 clicar <a href=\"http:\/\/get.adobe.com\/br\/reader\/\" rel=\"noopener\" target=\"_blank\"><strong>AQUI<\/strong><\/a> para fazer o download adequado ao seu dispositivo.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div style=\" float:right;  width: 100px;\">\n<p style=\"text-align: center; border-style: dotted; border-color :#FFFAFA; border-width :thin; width:90px;  height:40px;  padding: 5px; background-color: #800000;\">\n<a href=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2020\/11\/\u00daltimo-d\u00edgito.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ \u00daltimo d\u00edgito.pdf\"><span style=\"color: #FFFAFA; font-family: georgia; font-size: 16px;\"><strong>Arquivo <\/strong><\/span><\/a>\n <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Problema (A partir do 9\u00ba ano do E. F.- N\u00edvel de dificuldade: M\u00e9dio) Determine o algarismo das unidades da soma [tex] \\boxed{3^{2020}+4^{2020}}\\,[\/tex]. Adaptado do 6\u00ba Campeonato de Matem\u00e1tica de la Universidad de La Frontera, 2013. Ajuda Algoritmo de Euclides ou Divis\u00e3o Euclidiana Sejam [tex]a[\/tex] e [tex]b[\/tex] n\u00fameros naturais, com [tex]b\\ne 0.[\/tex] [tex]\\qquad \\qquad \\begin{array}{r} a &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-ultimo-digito\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-166957","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/166957","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=166957"}],"version-history":[{"count":16,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/166957\/revisions"}],"predecessor-version":[{"id":197837,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/166957\/revisions\/197837"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=166957"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}