{"id":143514,"date":"2019-12-06T01:46:12","date_gmt":"2019-12-06T03:46:12","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=143514"},"modified":"2024-06-23T06:42:45","modified_gmt":"2024-06-23T09:42:45","slug":"problema-para-ajudar-na-escola-numeros-equilibrados","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-numeros-equilibrados\/","title":{"rendered":".Problema para ajudar na escola: N\u00fameros equilibrados"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=220650\" rel=\"noopener\" target=\"_blank\">&#9999; <span style=\"color:#A0522D;font-size:14px;\">Link do problema para dispositivos da <em>Apple<\/em>.<\/span><\/a><\/strong><\/p>\n<div class=\"g1n\" style=\"line-height:28px;\">\n<p class=\"tit\" style=\"line-height:20px; margin-bottom: 10px;\">Problema<br \/>\n<span style=\"font-size:12px;color:#FF8C00\"> (A partir do 7\u00ba ano do E. F. &#8211; <span style=\"font-weight: normal;\"> N\u00edvel de dificuldade: Dif\u00edcil<\/span>)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>Um n\u00famero com tr\u00eas algarismos \u00e9 dito <strong><em>equilibrado <\/em><\/strong> se seu algarismo central for a soma dos outros dois.<br \/>\n Por exemplo, <\/p>\n<ul>\n<li>[tex]275 [\/tex] \u00e9 um n\u00famero equilibrado, j\u00e1 que [tex]2+5=7[\/tex];<\/li>\n<li>  [tex]846 [\/tex] n\u00e3o \u00e9  equilibrado, pois [tex]8+6\\ne 4[\/tex].<\/li>\n<\/ul>\n<p><strong>Quantos m\u00faltiplos de [tex]7[\/tex] com tr\u00eas algarismos s\u00e3o equilibrados?<\/strong><\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\n<p class=\"tit\" style=\"line-height:30px; margin-bottom: 10px;\">Solu\u00e7\u00e3o <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nNeste problema, vamos trabalhar com n\u00fameros de tr\u00eas algarismos cuja soma dos algarismos laterais  <span style=\"font-size: 12px;\">(algarismos da centena e da unidade)<\/span> \u00e9 igual ao algarismo central <span style=\"font-size: 12px;\">(algarismo da dezena)<\/span>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2019\/12\/prob390.png\" style=\"border:none\"  alt=\"\" width=\"209\" height=\"176\" class=\"aligncenter size-full wp-image-143461\" \/><\/p>\n<p>Ent\u00e3o, seja [tex]E[\/tex] um n\u00famero de tr\u00eas algarismos, equilibrado e m\u00faltiplo de [tex]7[\/tex], conforme exige o problema.<\/p>\n<ul>\n<li> Como [tex]E[\/tex] tem tr\u00eas algarismos, ent\u00e3o podemos escrever que:<\/li>\n<p>[tex]\\qquad E=rst=r\\,10^2+s\\,10^1+t=r \\cdot 100 +s \\cdot 10+t.\\qquad \\qquad \\textcolor{#800000}{(i)}[\/tex]<br \/>\n<span style=\"font-size:12px;\"> (Observe que aqui a nota\u00e7\u00e3o  [tex]rst[\/tex] n\u00e3o indica um produto e sim a representa\u00e7\u00e3o de um n\u00famero com tr\u00eas algarismos no sistema decimal: [tex]r[\/tex] \u00e9 o algarismo das centenas, [tex]s[\/tex] \u00e9 o algarismos das dezenas e [tex]t[\/tex] \u00e9 o algarismos das unidades.)<\/span><\/p>\n<li>Sendo [tex]E=rst[\/tex] um n\u00famero equilibrado, ent\u00e3o <\/li>\n<p>[tex]\\qquad s=r+t.\\qquad \\qquad \\textcolor{#800000}{(ii)}[\/tex]<\/ul>\n<p>Dessa forma, por [tex]\\textcolor{#800000}{(i)}[\/tex] e [tex]\\textcolor{#800000}{(ii)}[\/tex], segue que:<br \/>\n[tex]\\qquad E=r \\cdot 100 +s \\cdot 10+t[\/tex]<br \/>\n[tex]\\qquad E=r \\cdot 100 +(r+t) \\cdot 10+t[\/tex]<br \/>\n[tex]\\qquad E=r \\cdot 100 +r \\cdot 10+t \\cdot 10 +t[\/tex]<br \/>\n[tex]\\qquad E=r \\cdot 110 +t \\cdot 11[\/tex]<br \/>\n[tex]\\qquad E=11 \\cdot (r \\cdot 10 +t).\\qquad \\qquad \\textcolor{#800000}{(iii)}[\/tex]<br \/>\nComo [tex]r, s[\/tex]  e [tex]t[\/tex]  s\u00e3o algarismos, consequentemente s\u00e3o n\u00fameros naturais. Assim, [tex]n= r\\cdot 10 +t[\/tex] \u00e9 um n\u00famero natural e, portanto, segue de [tex]\\textcolor{#800000}{(iii)}[\/tex] que<br \/>\n[tex]\\qquad E=11 \\cdot n [\/tex], com [tex]n \\in \\mathbb{N}. \\qquad \\qquad  \\textcolor{#800000}{(iv)}[\/tex]<br \/>\n<span style=\"font-size:12px;\">Perceba que a conclus\u00e3o [tex]\\textcolor{#800000}{(iv)}[\/tex] s\u00f3 indica formalmente que [tex]E[\/tex] \u00e9 um m\u00faltiplo de [tex]11.[\/tex]<\/span><br \/>\nMas temos ainda a informa\u00e7\u00e3o de que [tex]E[\/tex] \u00e9 um m\u00faltiplo de [tex]7[\/tex]. Assim, [tex]7[\/tex] \u00e9 um divisor de [tex]E[\/tex] e, portanto, [tex]7[\/tex] deve aparecer na decomposi\u00e7\u00e3o de [tex]E[\/tex] como produto de fatores primos. Com essa informa\u00e7\u00e3o, podemos melhorar a decomposi\u00e7\u00e3o de [tex]E[\/tex]  e escrever a partir de [tex]\\textcolor{#800000}{(iv)}[\/tex] que:<br \/>\n[tex]\\qquad E=7 \\cdot 11 \\cdot m [\/tex], com [tex]m \\in \\mathbb{N}[\/tex]<br \/>\n[tex]\\qquad E=77 \\cdot m [\/tex], com [tex]m \\in \\mathbb{N}. \\qquad \\qquad  \\textcolor{#800000}{(v)}[\/tex]<br \/>\nComo [tex]\\boxed{77\\times 1=77}[\/tex], [tex]77\\times 2=144[\/tex], [tex]\\cdots[\/tex], [tex]77\\times 12=924[\/tex], [tex]\\boxed{77\\times 13=1001}[\/tex] e sabemos que [tex]E[\/tex] tem tr\u00eas algarismos, temos [tex]12-1=11[\/tex] possibilidades para que [tex]E=rst[\/tex] seja equilibrado. Vamos testar todas elas para verificar em quais delas temos [tex]\\boxed{s=r+t}[\/tex].<\/p>\n<table style=\"width:95%;border: 1px solid #ffffff; height:auto; margin-left: auto; margin-right:auto;\">\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]m[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=77 \\cdot m[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">Soma dos algarismos laterais<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">O algarismo central \u00e9 soma dos laterais?<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]2[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=154[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]1+4=5[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">SIM<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]3[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=231[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]2+1=3[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">SIM<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]4[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=308[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]3+8=11[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">N\u00c3O<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]5[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=385[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]3+5=8[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">SIM<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]6[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=462[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]4+2=6[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">SIM<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]7[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=539[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]5+9=14[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">N\u00c3O<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]8[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=616[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]6+6=12[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">N\u00c3O<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]9[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=693[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]6+3=9[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">SIM<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]10[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=770[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]7+0=7[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">SIM<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]11[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=847[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]8+7=15[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">N\u00c3O<\/td>\n<\/tr>\n<tr>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]12[\/tex]<\/td>\n<td class=\"bf\"style=\" border: 2px solid #aaaaaa;text-align:center\">[tex]E=924[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">[tex]9+4=13[\/tex]<\/td>\n<td class=\"bf\" style=\"border: 2px solid #aaaaaa;text-align:center\">N\u00c3O<\/td>\n<\/tr>\n<\/table>\n<p>A tabela acima nos mostra que existem [tex]\\fcolorbox{black}{#eee0e5}{$\\text{seis}$}[\/tex] m\u00faltiplos de [tex]7[\/tex] equilibrados com tr\u00eas algarismos:<br \/>\n<span style=\"color:#800000;\\padding-left:60px\">&#9658;<\/span> [tex]154,\\,231,\\,385,\\, 462,\\, 693,\\, 770 [\/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\/2019\/12\/N\u00fameros-equilibrados.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ N\u00fameros equilibrados.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>&#9999; Link do problema para dispositivos da Apple. Problema (A partir do 7\u00ba ano do E. F. &#8211; N\u00edvel de dificuldade: Dif\u00edcil) Um n\u00famero com tr\u00eas algarismos \u00e9 dito equilibrado se seu algarismo central for a soma dos outros dois. Por exemplo, [tex]275 [\/tex] \u00e9 um n\u00famero equilibrado, j\u00e1 que [tex]2+5=7[\/tex]; [tex]846 [\/tex] n\u00e3o \u00e9 &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-numeros-equilibrados\/\">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-143514","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/143514","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=143514"}],"version-history":[{"count":19,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/143514\/revisions"}],"predecessor-version":[{"id":220652,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/143514\/revisions\/220652"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=143514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}