{"id":52801,"date":"2015-12-03T00:00:42","date_gmt":"2015-12-03T02:00:42","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=52801"},"modified":"2020-09-22T22:13:45","modified_gmt":"2020-09-23T01:13:45","slug":"um-pouco-sobre-divisibilidade-tfa-problemas","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/teoria-dos-numeros-um-pouco-sobre-divisibilidade-parte-2\/um-pouco-sobre-divisibilidade-tfa-problemas\/","title":{"rendered":"Um pouco sobre divisibilidade: TFA &#8211; problemas"},"content":{"rendered":"<p style=\"margin:30px;\">\n<div class=\"grie cb22\">\n<p class=\"ti1 c2\" style=\"margin:10px;line-height:20px\"><em> Teorema Fundamental da Aritm\u00e9tica<\/em><\/p>\n<p class=\"ti2 c2\" style=\"margin:10px;line-height:20px\"><em> Problemas<\/em><\/p>\n<hr style=\"height:3px; width:100%;background-color:#4178a1; margin-top: 20px; margin-bottom: 50px;border: 0 none\"\/>\n<div class=\"gree cb24\" >\n<strong>Problema 1:<\/strong>  O n\u00famero [tex]2^4 \\cdot 5[\/tex] \u00e9 divis\u00edvel por [tex]2[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:10px 10px 10px 40px;\">\n Sim, j\u00e1 que [tex]2[\/tex] \u00e9 um dos fatores na decomposi\u00e7\u00e3o do n\u00famero dado.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 2:<\/strong>  O n\u00famero [tex]2^4\\cdot 5[\/tex] \u00e9 divis\u00edvel por [tex]3[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:10px 10px 10px 40px;\">\nN\u00e3o, j\u00e1 que a decomposi\u00e7\u00e3o deste n\u00famero n\u00e3o cont\u00e9m o n\u00famero primo [tex]3[\/tex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 3:<\/strong>  O n\u00famero [tex]2^4\\cdot 5 [\/tex] \u00e9 divis\u00edvel por [tex]8[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:10px 10px 10px 40px;\">\nSim, j\u00e1 que [tex]8=2^3[\/tex] e existem quatro fatores iguais a [tex]2[\/tex] na decomposi\u00e7\u00e3o do n\u00famero dado.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 4:<\/strong> O n\u00famero [tex]2^4\\cdot 5 [\/tex] \u00e9 divis\u00edvel por [tex]25[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:10px 10px 10px 40px;\">\nN\u00e3o, j\u00e1 que [tex]25=5^2[\/tex] e s\u00f3 existe um [tex]5[\/tex] na decomposi\u00e7\u00e3o do n\u00famero dado.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/12\/tfa1.png\" style=\"border:0px;\" alt=\"tfa1\" width=\"176\" height=\"167\" class=\"aligncenter size-full wp-image-52834\" \/>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 5:<\/strong> O n\u00famero n\u00famero [tex]2^4\\cdot 5 [\/tex] \u00e9 divis\u00edvel por [tex]10[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:10px 10px 10px 40px;\">\nSim, j\u00e1 que [tex]10=2\\cdot5[\/tex] e a decomposi\u00e7\u00e3o do n\u00famero dado cont\u00e9m ambos os n\u00fameros primos [tex]2[\/tex] e [tex]5[\/tex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/12\/tfa2.png\" style=\"border:0px;\" alt=\"tfa2\" width=\"125\" height=\"142\" class=\"aligncenter size-full wp-image-52835\" \/>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 6:<\/strong> O n\u00famero [tex]A[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]5[\/tex]. \u00c9 poss\u00edvel que o n\u00famero [tex]2A[\/tex] seja divis\u00edvel por [tex]5[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\nN\u00e3o, pois para que [tex]2A[\/tex] seja divis\u00edvel por [tex]5[\/tex] \u00e9 necess\u00e1rio e suficiente que na decomposi\u00e7\u00e3o do n\u00famero [tex]2A[\/tex]apare\u00e7a o fator [tex]5[\/tex]. Como esse fator n\u00e3o se encontra na fatora\u00e7\u00e3o do n\u00famero [tex]2[\/tex] nem na fatora\u00e7\u00e3o do n\u00famero [tex]A[\/tex] (afirmado pelo enunciado), podemos afirmar que [tex]2A[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]5[\/tex]\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 7:<\/strong> O n\u00famero [tex]A[\/tex] \u00e9 par. \u00c9 verdade que o n\u00famero [tex]2A[\/tex] tem que ser divis\u00edvel por  [tex]4[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\nInicialmente, lembramos que [tex]4=2^2[\/tex]. A resposta ao problema \u00e9 sim, pois observe que, se [tex]A[\/tex] \u00e9 par, ent\u00e3o [tex]A = 2n[\/tex], para algum n\u00famero natural [tex]n[\/tex].<br \/>\nPortanto, [tex]2A = 2\\cdot 2n[\/tex], ou seja, conclu\u00edmos que h\u00e1 pelo menos dois fatores [tex]2[\/tex] na decomposi\u00e7\u00e3o do n\u00famero [tex]2A[\/tex].\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 8:<\/strong> O n\u00famero [tex]3A[\/tex] \u00e9 divis\u00edvel por [tex]7[\/tex]. \u00c9 verdade que o n\u00famero [tex]A[\/tex] tem que ser divis\u00edvel por [tex]7[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:10px 10px 10px 40px;\">Sim, pois a decomposi\u00e7\u00e3o de [tex]3A[\/tex] cont\u00e9m [tex]7[\/tex], mas a decomposi\u00e7\u00e3o de [tex]3[\/tex] n\u00e3o.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 9:<\/strong> O n\u00famero [tex]21A[\/tex] \u00e9 divis\u00edvel por [tex]6[\/tex]. \u00c9 verdade que [tex]A[\/tex] tem que ser divis\u00edvel por [tex]6[\/tex]?<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\nN\u00e3o. Por exemplo, [tex]A[\/tex] poderia ser igual a [tex]2[\/tex]. Como [tex]6 = 2\\cdot 3[\/tex], ent\u00e3o para que um n\u00famero seja divis\u00edvel por [tex]6[\/tex] \u00e9 necess\u00e1rio que este tenha, em sua fatora\u00e7\u00e3o, os fatores [tex]2[\/tex] e [tex]3[\/tex]. Como [tex]21 = 3\\cdot 7[\/tex], ent\u00e3o podemos ter a certeza de que o n\u00famero [tex]A[\/tex] possui um fator [tex]2[\/tex] em sua decomposi\u00e7\u00e3o, mas n\u00e3o \u00e9 necess\u00e1ria a presen\u00e7a do fator [tex]3[\/tex]. Portanto, n\u00e3o \u00e9 preciso que A seja divis\u00edvel por [tex]6[\/tex].\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 10:<\/strong> Encontre o menor n\u00famero natural [tex]n[\/tex] tal que [tex]n![\/tex] \u00e9 divis\u00edvel por [tex]990[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo2 c2 cb23\" value=\"Resposta\" title=\"Clique aqui\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Resposta';}\" \/>    <\/p>\n<div  style=\"display: none;\">\nComo [tex]990=2\\cdot3^2\\cdot5\\cdot11[\/tex], para que [tex]n![\/tex] seja divis\u00edvel por [tex]990[\/tex], \u00e9 necess\u00e1rio que em sua decomposi\u00e7\u00e3o haja todos esses fatores.<br \/>\nMas [tex]11[\/tex] \u00e9 primo, logo ele mesmo tem que estar contido no produto.<br \/>\nObserve que [tex]11![\/tex] \u00e9 divis\u00edvel por [tex]2[\/tex], por [tex]3^2[\/tex] e por [tex]5[\/tex].<br \/>\nAssim, [tex]n=11[\/tex] \u00e9 o menor valor poss\u00edvel, pois o fatorial de qualquer outro n\u00famero menor que este n\u00e3o ter\u00e1 o fator [tex]11[\/tex] em sua decomposi\u00e7\u00e3o.\n<\/div>\n<\/div>\n<\/div>\n<p class=\"ti2 c2\"style=\"margin:30px 100px 30px 100px;\"><strong> Os pr\u00f3ximos problemas s\u00e3o para voc\u00eas fazerem!<\/strong><\/p>\n<div class=\"gree cb24\" >\n<p><strong>Problema 11:<\/strong> O n\u00famero [tex]5A[\/tex] \u00e9 divis\u00edvel por [tex]3[\/tex]. \u00c9 verdade que [tex]A[\/tex] tem que ser divis\u00edvel por [tex]3[\/tex]?\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 12:<\/strong> Encontre todos os n\u00fameros naturais [tex]x[\/tex] e [tex]y[\/tex] tais que [tex]x^2-y^2=31[\/tex]. (Resposta: Apenas [tex]x=16[\/tex] e [tex]y=15[\/tex])\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" >\n<p><strong>Problema 13:<\/strong> (OBM 2010 \u2013 F1N2) Qual das alternativas apresenta um divisor de [tex]3^5 \\times 4^4 \\times 5^3[\/tex] ?<br \/>\na) [tex]42\\qquad [\/tex] b) [tex]45\\qquad [\/tex] c) [tex]52\\qquad [\/tex] d) [tex]85\\qquad [\/tex] e) [tex]105\\qquad [\/tex](Resposta: b)\n<\/div>\n<hr style=\"height:2px; width:100%;background-color:#4178a1; margin-top: 40px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:2px; width:100%;background-color:#4178a1; margin-top: 8px; margin-bottom: 20px;border: 0 none\"\/>\n<p style=\"text-align: right;\"><span style=\"color: #4178a1; font-size: 18px;\"><strong>Equipe COM &#8211; OBMEP<\/strong><\/span><\/p>\n<\/div>\n<p class=\"fim\"><a title=\"Voltar.\" href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=52359\" target=\"_blank\"><span style=\"color: #6495ED;\"><strong>Voltar para Sala Principal do TFA<\/strong><\/span><\/a><\/p>\n<p style=\"margin:30px;\">\n","protected":false},"excerpt":{"rendered":"<p>Teorema Fundamental da Aritm\u00e9tica Problemas Problema 1: O n\u00famero [tex]2^4 \\cdot 5[\/tex] \u00e9 divis\u00edvel por [tex]2[\/tex]? Sim, j\u00e1 que [tex]2[\/tex] \u00e9 um dos fatores na decomposi\u00e7\u00e3o do n\u00famero dado. Problema 2: O n\u00famero [tex]2^4\\cdot 5[\/tex] \u00e9 divis\u00edvel por [tex]3[\/tex]? N\u00e3o, j\u00e1 que a decomposi\u00e7\u00e3o deste n\u00famero n\u00e3o cont\u00e9m o n\u00famero primo [tex]3[\/tex]. Problema 3: O &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/teoria-dos-numeros-um-pouco-sobre-divisibilidade-parte-2\/um-pouco-sobre-divisibilidade-tfa-problemas\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":2970,"parent":51902,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-52801","page","type-page","status-publish","has-post-thumbnail","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/52801","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=52801"}],"version-history":[{"count":22,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/52801\/revisions"}],"predecessor-version":[{"id":165329,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/52801\/revisions\/165329"}],"up":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/51902"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media\/2970"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=52801"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}