{"id":150278,"date":"2020-04-07T01:28:12","date_gmt":"2020-04-07T04:28:12","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=150278"},"modified":"2023-06-11T00:55:51","modified_gmt":"2023-06-11T03:55:51","slug":"problema-divisao-euclidiana","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-divisao-euclidiana\/","title":{"rendered":".Problema: Divis\u00e3o Euclidiana"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:25px;\">\n<p class=\"tit\" style=\"line-height:20px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 7\u00ba ano do E. F.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>Na divis\u00e3o euclidiana de dois inteiros positivos [tex]a[\/tex] e [tex]b[\/tex], com [tex]a\\gt b[\/tex], o quociente \u00e9  [tex]16[\/tex] e o resto \u00e9  o maior poss\u00edvel. Encontrar  [tex]a[\/tex] e [tex]b[\/tex], sabendo que [tex]a+b=341[\/tex].<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"cb31 g1n\" style=\"margin:30px;line-height:25px; width:680px; 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\">\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=q \\, b+r.[\/tex]<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\" style=\"line-height:20px;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\"\/>\n<p>Pelo processo de <strong>Divis\u00e3o Euclidiana<\/strong>, a partir das hip\u00f3teses do problema podemos concluir que existe um inteiro [tex]r[\/tex] de modo que [tex]\\boxed{a= 16\\cdot b+r}[\/tex], com [tex]0\\leq r \\lt b[\/tex].<br \/>\nComo [tex]a+b=341[\/tex], vem que [tex]\\boxed{a=341-b}[\/tex] e, ent\u00e3o, segue que:<br \/>\n[tex]\\qquad 341-b=16\\cdot b+r[\/tex]<br \/>\n[tex]\\qquad 17b+r=341\\,. \\qquad \\qquad \\textcolor{#800000}{(*)}[\/tex]<br \/>\nComo [tex]r[\/tex] deve ser o maior valor poss\u00edvel temos que [tex]\\boxed{r=b-1}[\/tex]. Assim, por [tex] \\textcolor{#800000}{(*)}[\/tex]:<br \/>\n[tex]\\qquad 17b+(b-1)=341[\/tex]<br \/>\n[tex]\\qquad 18b=342[\/tex]<br \/>\n[tex]\\qquad \\, \\fcolorbox{black}{#eee0e5}{$b=19$}\\,.[\/tex]<br \/>\nPortanto, [tex]a=341-19[\/tex], ou seja, [tex]\\, \\fcolorbox{black}{#eee0e5}{$a=322$}\\,.[\/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:10px\">\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 7\u00ba ano do E. F.) Na divis\u00e3o euclidiana de dois inteiros positivos [tex]a[\/tex] e [tex]b[\/tex], com [tex]a\\gt b[\/tex], o quociente \u00e9 [tex]16[\/tex] e o resto \u00e9 o maior poss\u00edvel. Encontrar [tex]a[\/tex] e [tex]b[\/tex], sabendo que [tex]a+b=341[\/tex]. Ajuda Algoritmo de Euclides ou Divis\u00e3o Euclidiana Sejam [tex]a[\/tex] e [tex]b[\/tex] n\u00fameros naturais, com &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-divisao-euclidiana\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":721,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-150278","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/150278","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\/721"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=150278"}],"version-history":[{"count":7,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/150278\/revisions"}],"predecessor-version":[{"id":194273,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/150278\/revisions\/194273"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=150278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}