{"id":11227,"date":"2012-11-11T22:41:21","date_gmt":"2012-11-11T22:41:21","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11227"},"modified":"2022-01-02T13:34:49","modified_gmt":"2022-01-02T13:34:49","slug":"o-numero-a","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11227","title":{"rendered":"O n\u00famero $A$"},"content":{"rendered":"<p><ul id='GTTabs_ul_11227' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11227' class='GTTabs_curr'><a  id=\"11227_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11227' ><a  id=\"11227_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11227'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>O n\u00famero $A = {3^2} \\times 7 \\times 11$ n\u00e3o \u00e9 um n\u00famero quadrado perfeito.<\/p>\n<p>Qual o menor n\u00famero inteiro pelo qual devemos multiplicar $A$ para obtemos um quadrado perfeito?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11227' onClick='GTTabs_show(1,11227)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11227'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Comecemos por um n\u00famero mais pequeno para vermos o que se passa. Consideremos, por exemplo, $B = {3^2} \\times 2 = 18$.<\/p>\n<p>Ora, $B$ n\u00e3o \u00e9 um quadrado perfeito. O pr\u00f3ximos quadrados perfeitos s\u00e3o: $25$, $36$, $49$, &#8230;<\/p>\n<p>O n\u00famero $25$ n\u00e3o \u00e9 m\u00faltiplo de $B$, mas j\u00e1 $36$ \u00e9 o menor m\u00faltiplo de $B$ que \u00e9 superior a ele.<\/p>\n<p>Multipliquemos o n\u00famero $B$ por $2$, para vermos o que acontece:<\/p>\n<p>$$2 \\times B = 2 \\times \\left( {{3^2} \\times 2} \\right) = {3^2} \\times {2^2} = \\underbrace {{{\\left( {3 \\times 2} \\right)}^2} = 36}_{{\\text{quadrado perfeito}}}$$<\/p>\n<\/p>\n<p>Consideremos, agora, o n\u00famero dado: $A = {3^2} \\times 7 \\times 11$.<\/p>\n<p>Ora, o menor n\u00famero inteiro pelo qual devemos multiplicar $A$ para obtermos um quadrado perfeito \u00e9 $N = 7 \\times 11 = 77$.<\/p>\n<p>Com efeito, vem:<\/p>\n<p>$$77 \\times A = 7 \\times 11 \\times \\left( {{3^2} \\times 7 \\times 11} \\right) = {3^2} \\times {7^2} \\times {11^2} = \\underbrace {{{\\left( {3 \\times 7 \\times 11} \\right)}^2}}_{{\\text{quadrado perfeito}}}$$<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11227' onClick='GTTabs_show(0,11227)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O n\u00famero $A = {3^2} \\times 7 \\times 11$ n\u00e3o \u00e9 um n\u00famero quadrado perfeito. Qual o menor n\u00famero inteiro pelo qual devemos multiplicar $A$ para obtemos um quadrado perfeito? Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19290,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[317,97,318],"tags":[319,340,339],"series":[],"class_list":["post-11227","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-7-o-ano","category-aplicando","category-numeros-inteiros","tag-numeros-inteiros-2","tag-quadrado-perfeito","tag-raiz-quadrada"],"views":2643,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat108.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11227","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11227"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11227\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19290"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11227"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}