{"id":129209,"date":"2019-06-05T19:36:49","date_gmt":"2019-06-05T22:36:49","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=129209"},"modified":"2024-06-10T22:03:44","modified_gmt":"2024-06-11T01:03:44","slug":"problema-para-ajudar-na-escola-uma-soma-de-quadrados","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-uma-soma-de-quadrados\/","title":{"rendered":".Problema para ajudar na escola:  Uma soma de quadrados"},"content":{"rendered":"<p class=\"g2\" ><strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=220227\" 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:25px;\">\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> Determinar de quantas maneiras podemos escrever [tex]2003 [\/tex] como soma dos quadrados de dois n\u00fameros naturais n\u00e3o nulos. <\/p>\n<p style=\"margin:20px\">\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"cb36 g1\" style=\"margin:30px;line-height:30px; 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=\"99\" height=\"120\" class=\"alignleft size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\" style=\"line-height:20px; margin-bottom: 10px;\"> Ajuda <\/p>\n<p>Em v\u00e1rias situa\u00e7\u00f5es \u00e9 poss\u00edvel determinarmos a paridade de express\u00f5es envolvendo n\u00fameros naturais, a partir da paridade desses n\u00fameros, sem sequer calcular as express\u00f5es.<br \/>\nParticularmente, as tabelinhas abaixo ilustram situa\u00e7\u00f5es de paridade que ir\u00e3o aparecer na solu\u00e7\u00e3o deste problema:<\/p>\n<ul>\n<li>A soma de dois n\u00fameros naturais de mesma paridade \u00e9 par.<\/li>\n<li>A soma de dois n\u00fameros naturais de paridade oposta \u00e9 \u00edmpar.<\/li>\n<li>O produto de dois n\u00fameros naturais s\u00f3 ser\u00e1 \u00edmpar se os dois n\u00fameros forem \u00edmpares.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2016\/01\/aritmetica.png\" alt=\"aritmetica\" width=\"625\" height=\"106\" class=\"aligncenter size-full wp-image-54884\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2016\/01\/aritmetica.png 625w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2016\/01\/aritmetica-300x51.png 300w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><\/p>\n<p style=\"font-size:12px;text-align:right;\">(Se precisar, visite <a title=\"Paridade da soma e do produto\" href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=54839\" target=\"_blank\"><span class=\"c3\"><strong> esta Sala<\/strong><\/span><\/a>.)<\/p>\n<\/div>\n<p style=\"margin:40px\">\n<div class=\"g1n\" style=\"line-height:27px;\">\n<p class=\"tit\" style=\"line-height:30px; margin-bottom: 10px;\">Solu\u00e7\u00e3o 1<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 30px;border: 0 none\"\/>\n<p>Sejam [tex]x \\, [\/tex] e [tex] \\, y[\/tex] n\u00fameros naturais n\u00e3o nulos que satisfa\u00e7am as condi\u00e7\u00f5es do problema. Assim, [tex]x^2+y^2=2003.[\/tex]<\/p>\n<ul>\n<li>Como [tex]2003[\/tex] \u00e9 um n\u00famero \u00edmpar, ent\u00e3o as propriedades elencadas na <strong>Ajuda <\/strong>apontam que [tex]x^2[\/tex] e [tex]y^2[\/tex] t\u00eam paridades opostas (veja a tabelinha da esquerda).<\/li>\n<li>Mas a tabelinha da direita mostra que o quadrado de um n\u00famero mant\u00e9m a sua paridade; assim, [tex]x \\, [\/tex] e [tex] \\, y[\/tex] t\u00eam paridades contr\u00e1rias.<\/li>\n<\/ul>\n<p>Vamos supor, sem perda de generalidade, que [tex]x[\/tex] seja um n\u00famero par e [tex] \\, y[\/tex] seja um n\u00famero \u00edmpar. Dessa forma, existem n\u00fameros naturais [tex]m \\, [\/tex] e [tex] \\, n \\, [\/tex], com  [tex] \\, n\\ne 0[\/tex], tais que [tex] \\, \\boxed{x=2n} \\, [\/tex] e [tex] \\, \\boxed{y=2m+1}[\/tex], donde segue que:<br \/>\n[tex]\\qquad x^2+y^2=2003[\/tex]<br \/>\n[tex]\\qquad (2n)^2+(2m+1)^2=2003[\/tex]<br \/>\n[tex]\\qquad 4n^2+4m^2+4m+1=2003[\/tex]<br \/>\n[tex]\\qquad 4\\left(n^2+m^2+m\\right)+1=2003[\/tex]<br \/>\n[tex]\\qquad 4\\left(n^2+m^2+m\\right)=2002. \\qquad \\textcolor{#800000}{(i)}[\/tex]<br \/>\nPerceba que se [tex]k=n^2+m^2+m[\/tex], ent\u00e3o de [tex]\\textcolor{#800000}{(i)}[\/tex] segue que [tex]2002=4k[\/tex], com [tex]k\\in \\mathbb{N}[\/tex] e com isso concluir\u00edamos que [tex]2002[\/tex] \u00e9 um m\u00faltiplo de [tex]4.[\/tex] Mas<br \/>\n[tex]\\qquad \\qquad \\begin{array}{r}<br \/>\n2002 \\, \\end{array}   \\begin{array}{|r}<br \/>\n \\, 4 \\,  \\,  \\,  \\,  \\\\  \\hline<br \/>\n\\end{array}[\/tex]<br \/>\n[tex]\\quad \\qquad\\begin{array}{r}<br \/>\n\\qquad \\textcolor{red}{2}<br \/>\n\\end{array}\\begin{array}{r}<br \/>\n\\quad   500<br \/>\n\\end{array}[\/tex]<br \/>\no que mostra que [tex]2002[\/tex] N\u00c3O \u00e9 um m\u00faltiplo de [tex]4.[\/tex]<br \/>\nPortanto, n\u00e3o podemos supor que existam n\u00fameros naturais n\u00e3o nulos cuja soma dos quadrados seja igual a [tex]2003[\/tex], pois isso acarreta um erro.<\/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:40px\">\n<table style=\" height:auto; margin-left: auto; margin-right: auto; width:533px;\">\n<tr>\n<td style=\"border: 1px solid #ffffff\">\n \t \t<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=\"95\" height=\"115\" \/><\/td>\n<td style=\"border: 1px solid #ffffff\">\n<p class=\"cb33 bg32\" style=\"text-align: justify; width: 370px; padding: 10px;\"><span style=\"color: #1c1c1c; font-family: calibri, sans-serif; font-size: 16px;\">Se voc\u00ea ainda n\u00e3o aprendeu a lidar com a \u00e1lgebra, n\u00e3o faz mal, veja a segunda solu\u00e7\u00e3o.<\/span><\/p>\n<p><\/td>\n<\/tr>\n<\/table>\n<div class=\"g1n\" style=\"line-height:25px;\">\n<p class=\"tit\" style=\"line-height:30px; margin-bottom: 10px;\">Solu\u00e7\u00e3o 2<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 30px;border: 0 none\"\/>\n<p>Os esquemas multiplicativos abaixo nos mostram que o quadrado de um n\u00famero natural n\u00e3o nulo termina necessariamente em [tex]0,1,4,5,6[\/tex] ou [tex]9.[\/tex]<\/p>\n<table style=\"border: 1px solid #ffffff; width:95%; height:auto; text-align:center;  margin-left: auto; margin-right:auto;font-size:13px\">\n<tr class=\"bf\">\n<td style=\"border: 1px solid #ffffff; \">\n[tex]\\begin{array}{c c r c}<br \/>\n \\,  \\, \\\\<br \/>\n&#038;a&#038;\\cdots&#038;1\\\\<br \/>\n\\times &#038;a&#038;\\cdots&#038;1\\\\<br \/>\n\\hline<br \/>\n&#038;b&#038;\\cdots&#038;\\boxed{1}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n \\,  \\, \\\\<br \/>\n&#038;a&#038;\\cdots&#038;2\\\\<br \/>\n\\times &#038;a&#038;\\cdots&#038;2\\\\<br \/>\n\\hline<br \/>\n&#038;c&#038;\\cdots&#038;\\boxed{4}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n \\,  \\, \\\\<br \/>\n&#038;a&#038;\\cdots&#038;3\\\\<br \/>\n\\times &#038;a&#038;\\cdots&#038;3\\\\<br \/>\n\\hline<br \/>\n&#038;d&#038;\\cdots&#038;\\boxed{9}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n&#038;&#038;_\\underline{1}&#038;\\\\<br \/>\n&#038;a&#038;\\cdots&#038;4\\\\<br \/>\n\\times&#038;a&#038;\\cdots&#038;4\\\\<br \/>\n\\hline<br \/>\n&#038;e&#038;\\cdots&#038;\\boxed{6}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<\/tr>\n<tr class=\"bf\">\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n&#038;&#038;_\\underline{2}&#038;\\\\<br \/>\n&#038;a&#038;\\cdots&#038;5\\\\<br \/>\n\\times&#038;a&#038;\\cdots&#038;5\\\\<br \/>\n\\hline<br \/>\n&#038;f&#038;\\cdots&#038;\\boxed{5}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n&#038;&#038;_\\underline{3}&#038;\\\\<br \/>\n&#038;a&#038;\\cdots&#038;6\\\\<br \/>\n\\times&#038;a&#038;\\cdots&#038;6\\\\<br \/>\n\\hline<br \/>\n&#038;g&#038;\\cdots&#038;\\boxed{6}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n&#038;&#038;_\\underline{4}&#038;\\\\<br \/>\n&#038;a&#038;\\cdots&#038;7\\\\<br \/>\n\\times&#038;a&#038;\\cdots&#038;7\\\\<br \/>\n\\hline<br \/>\n&#038;h&#038;\\cdots&#038;\\boxed{9}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n&#038;&#038;_\\underline{6}&#038;\\\\<br \/>\n&#038;a&#038;\\cdots&#038;8\\\\<br \/>\n\\times&#038;a&#038;\\cdots&#038;8\\\\<br \/>\n\\hline<br \/>\n&#038;l&#038;\\cdots&#038;\\boxed{4}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n&#038;&#038;_\\underline{8}&#038;\\\\<br \/>\n&#038;a&#038;\\cdots&#038;9\\\\<br \/>\n\\times&#038;a&#038;\\cdots&#038;9\\\\<br \/>\n\\hline<br \/>\n&#038;r&#038;\\cdots&#038;\\boxed{1}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<td style=\"border: 1px solid #ffffff; \">\n[tex]<br \/>\n\\begin{array}{c c r c}<br \/>\n&#038;&#038;&#038;\\\\<br \/>\n&#038;a&#038;\\cdots&#038;0\\\\<br \/>\n\\times&#038;a&#038;\\cdots&#038;0\\\\<br \/>\n\\hline<br \/>\n&#038;s&#038;\\cdots&#038;\\boxed{0}<br \/>\n\\end{array}<br \/>\n[\/tex]\n<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:10px\">\n<p>Dessa forma, para que a soma de dois quadrados seja [tex]2003[\/tex] ([tex]x^2+y^2=2003[\/tex]), um dos quadrados deve terminar em [tex]4[\/tex] e o outro em [tex]9.[\/tex]<br \/>\nPara que o quadrado de um n\u00famero natural termine em [tex]9[\/tex], esse n\u00famero deve terminar em [tex]3[\/tex] ou em [tex]7[\/tex], conforme mostram os esqueminhas acima; assim, podemos listar os quadrados terminados em [tex]9[\/tex] que s\u00e3o menores do que [tex]2003[\/tex], para verificar quais diferen\u00e7as [tex]2003-x^2[\/tex] (que sabemos que terminam em [tex]4[\/tex]) s\u00e3o quadrados perfeitos.<br \/>\nVamos l\u00e1!<\/p>\n<table style=\"border: 2px solid #616161; width:80%; height:auto; text-align:center; margin-left:auto; margin-right:auto;\">\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] x[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]x^2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]2003-x^2[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]2003-x^2[\/tex] \u00e9 quadrado perfeito?<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 3[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]9[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1994[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1994[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{1994} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 7[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]49[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1954[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1954[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{1954} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 13[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]169[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1834[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1834[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{1834} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 17[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]289[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1714[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1714[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{1714} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 23[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]529[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1474[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1474[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{1474} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 27[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]729[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1274[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1274[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{1274} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 33[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1089[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]914[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]914[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{914} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 37[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1369[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]634[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]634[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{634} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 2px solid #616161;\">[tex] 43[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]1849[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]154[\/tex]<\/td>\n<td style=\"border: 2px solid #616161;\">[tex]154[\/tex] n\u00e3o \u00e9 quadrado perfeito, pois [tex]\\sqrt{154} \\not\\in \\mathbb{N}[\/tex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"4\" style=\"border: 2px solid #616161;\">[tex] 47^2=2209 \\gt 2003[\/tex]<\/td>\n<\/tr>\n<\/table>\n<p style=\"margin:10px\">\n<p>Assim, n\u00e3o existem n\u00fameros naturais n\u00e3o nulos [tex]x \\, [\/tex] e [tex] \\, y[\/tex] tais que [tex]x^2+y^2=2003.[\/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\/Uma-soma-de-quadrados.pdf\" title=\" Arquivo PDF\" download=\"Clubes_ Uma soma de quadrados.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) Determinar de quantas maneiras podemos escrever [tex]2003 [\/tex] como soma dos quadrados de dois n\u00fameros naturais n\u00e3o nulos. Ajuda Em v\u00e1rias situa\u00e7\u00f5es \u00e9 poss\u00edvel determinarmos a paridade de express\u00f5es envolvendo n\u00fameros naturais, a &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problema-para-ajudar-na-escola-uma-soma-de-quadrados\/\">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-129209","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/129209","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=129209"}],"version-history":[{"count":33,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/129209\/revisions"}],"predecessor-version":[{"id":220229,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/129209\/revisions\/220229"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=129209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}