{"id":18388,"date":"2014-05-04T09:01:34","date_gmt":"2014-05-04T12:01:34","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=18388"},"modified":"2023-01-30T02:56:18","modified_gmt":"2023-01-30T05:56:18","slug":"probleminha-menor-numero-natural","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/probleminha-menor-numero-natural\/","title":{"rendered":".Probleminha: Um menor n\u00famero natural"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 6\u00ba ano do E. F.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 10px;border: 0 none\"\/>\n<p style=\"margin:20px\">\n<p>Qual o menor n\u00famero natural que<\/p>\n<ul>\n<li>\n<p>dividido por <span class=\"lat\">3<\/span> deixa resto <span class=\"lat\">2<\/span>,<\/p>\n<\/li>\n<li>\n<p>dividido por <span class=\"lat\">4<\/span> deixa resto <span class=\"lat\">3<\/span>,<\/p>\n<\/li>\n<li>\n<p>dividido por <span class=\"lat\">5<\/span> deixa resto <span class=\"lat\">4<\/span>,<\/p>\n<\/li>\n<li>\n<p>dividido por<span class=\"lat\">6<\/span> deixa resto <span class=\"lat\">5<\/span>,<\/p>\n<\/li>\n<li>\n<p>dividido por <span class=\"lat\">7<\/span> deixa resto <span class=\"lat\">6<\/span>,<\/p>\n<\/li>\n<li>\n<p>dividido por <span class=\"lat\">8<\/span> deixa resto <span class=\"lat\">7<\/span>,<\/p>\n<\/li>\n<li>\n<p>dividido por <span class=\"lat\">9<\/span> deixa resto <span class=\"lat\">8<\/span>,<\/p>\n<\/li>\n<li>\n<p>dividido por <span class=\"lat\">10<\/span> deixa resto <span class=\"lat\">9<\/span>?<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Solu\u00e7\u00e3o 1 <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n Se esse n\u00famero fosse divis\u00edvel por <span class=\"lat\">2, 3, 4, 5, 6, 7, 8, 9, 10<\/span>, ele seria o <span class=\"lat\">MMC(2, 3, 4, 5, 6, 7, 8, 9, 10)=2520<\/span>.<br \/>\nPor\u00e9m, observe que a diferen\u00e7a entre o divisor e o resto em todas as divis\u00f5es \u00e9  <span class=\"lat\">1<\/span>, desse modo, o menor n\u00famero natural que satisfaz a todas as condi\u00e7\u00f5es do problema \u00e9  <span class=\"lat\">2520 &#8211; 1 = 2519<\/span>.<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Solu\u00e7\u00e3o 2<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicada a partir do 8\u00ba ano do E. F.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 10px;border: 0 none\"\/>\n<p style=\"margin:20px\">\n<p>Seja  <span class=\"lati\">x<\/span> esse n\u00famero; assim, podemos reescrever as hip\u00f3teses respectivamente como:<\/p>\n<ul>\n<li><span class=\"lati\">x = 3p + 2<\/span>, <\/li>\n<li><span class=\"lati\">x = 4q + 3<\/span>,<\/li>\n<li><span class=\"lati\">x = 5r + 4<\/span>,<\/li>\n<li><span class=\"lati\">x = 6s + 5<\/span>, <\/li>\n<li><span class=\"lati\">x = 7t + 6<\/span>,<\/li>\n<li><span class=\"lati\">x = 8n + 7<\/span>, <\/li>\n<li><span class=\"lati\">x = 9m + 8<\/span>,<\/li>\n<li><span class=\"lati\">x = 10h + 9<\/span>.<\/li>\n<\/ul>\n<p>Dessa forma,<\/p>\n<ul>\n<li> <span class=\"lati\"> x + 1 = 3p + 3 = 3( p + 1 )<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 4q + 4 = 4( q + 1 )<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 5r + 5 = 5( r + 1 )<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 6s + 6 = 6( s + 1 )<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 7t + 7 = 7( t + 1 )<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 8n + 8 = 8( n + 1 )<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 9m + 9 = 9( m + 1 )<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 10h + 10 = 10( h + 1 )<\/span>,<\/li>\n<\/ul>\n<p>e, assim, <\/p>\n<ul>\n<li> <span class=\"lati\"> x + 1 = 3p<sub>1<\/sub>, p<sub>1<\/sub> &isin; N<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 4q<sub>1<\/sub>, q<sub>1<\/sub> &isin; N<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 5r<sub>1<\/sub>, r<sub>1<\/sub> &isin; N<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 6s<sub>1<\/sub>, s<sub>1<\/sub> &isin; N<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 7t<sub>1<\/sub>, t<sub>1<\/sub> &isin; N<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 8n<sub>1<\/sub>, n<sub>1<\/sub> &isin; N<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 9m<sub>1<\/sub>, m<sub>1<\/sub> &isin; N<\/span>,<\/li>\n<li> <span class=\"lati\"> x + 1 = 10h<sub>1<\/sub>, h<sub>1<\/sub> &isin; N<\/span>.<\/li>\n<\/ul>\n<p>Podemos concluir, ent\u00e3o, que <span class=\"lati\"> x + 1<\/span> \u00e9 simultaneamente m\u00faltiplo de <span class=\"lat\">2, 3, 4, 5, 6, 7, 8, 9, 10<\/span>.<br \/>\nComo queremos o menor valor de <span class=\"lati\"> x + 1<\/span>, ent\u00e3o  <span class=\"lati\"> x + 1<\/span> \u00e9 o m\u00ednimo m\u00faltiplo comum de <span class=\"lat\">2, 3, 4, 5, 6, 7, 8, 9, 10)<\/span>.<br \/>\nPortanto,<br \/>\n <span class=\"lat\" style=\"padding-left:30px;\"><em>x + 1<\/em> = MMC(2, 3, 4, 5, 6, 7, 8, 9, 10)=2520<\/span><br \/>\n e<br \/>\n<span class=\"lat\" style=\"padding-left:30px;\"><em>x<\/em> = 2519<\/span>. <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g3\"> Participaram da discuss\u00e3o os Clubes: <strong>Alfa e \u00d4mega; MIRIM APRENDIZ; Os Aprendizes de Einstein<\/strong>.<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 6\u00ba ano do E. F.) Qual o menor n\u00famero natural que dividido por 3 deixa resto 2, dividido por 4 deixa resto 3, dividido por 5 deixa resto 4, dividido por6 deixa resto 5, dividido por 7 deixa resto 6, dividido por 8 deixa resto 7, dividido por 9 deixa &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/probleminha-menor-numero-natural\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-18388","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/18388","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=18388"}],"version-history":[{"count":122,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/18388\/revisions"}],"predecessor-version":[{"id":183640,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/18388\/revisions\/183640"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=18388"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}