{"id":52717,"date":"2015-12-01T11:59:48","date_gmt":"2015-12-01T13:59:48","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=52717"},"modified":"2020-11-03T22:05:37","modified_gmt":"2020-11-04T00:05:37","slug":"um-pouco-sobre-divisibilidade-criterios-de-divisibilidade-sem-justificativas","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/um-pouco-sobre-divisibilidade-criterios-de-divisibilidade-sem-justificativas\/","title":{"rendered":"Um pouco sobre divisibilidade &#8211; Crit\u00e9rios de divisibilidade, sem  justificativas"},"content":{"rendered":"<p style=\"margin:30px;\">\n<div class=\"grie cb22\" style=\"width:80%\">\n<p class=\"ti1 c2\">  Crit\u00e9rios de divisibilidade<\/p>\n<hr style=\"height:3px;background-color:#4178a1; border: 0 none\"\/>\n<p style=\"margin:30px 10px 10px 10px;\">\nNesta Sala, apresentaremos alguns dos tradicionais crit\u00e9rios de divisibilidade.<br \/>\nFixado um n\u00famero natural n\u00e3o nulo [tex]d[\/tex], um <strong>crit\u00e9rio de divisibilidade<\/strong> \u00e9 uma condi\u00e7\u00e3o [tex]P[\/tex] necess\u00e1ria e suficiente para que um n\u00famero natural seja divis\u00edvel por [tex]d[\/tex], portanto algo do tipo:<\/p>\n<p class=\"ti3 c2\"> Um n\u00famero natural <em>n<\/em> \u00e9 divis\u00edvel por <em>d<\/em> se, e somente se <em>P<\/em>.<\/p>\n<p style=\"margin:30px 10px 10px 10px;\">Isso n\u00e3o s\u00f3 significa que significa que &#8220;se [tex]P[\/tex] ocorre, ent\u00e3o <em>n<\/em> \u00e9 divis\u00edvel por [tex]d[\/tex]&#8221;, mas tamb\u00e9m significa que &#8220;se a condi\u00e7\u00e3o [tex]P[\/tex] n\u00e3o ocorre, ent\u00e3o [tex]n[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]d[\/tex]&#8221;.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/11\/criterio.png\" style=\"border:0px\" alt=\"criterio\" width=\"544\" height=\"218\" class=\"aligncenter size-full wp-image-52021\" srcset=\"https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/11\/criterio.png 544w, https:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2015\/11\/criterio-300x120.png 300w\" sizes=\"auto, (max-width: 544px) 100vw, 544px\" \/><\/p>\n<p style=\"margin:20px;\">\n<hr style=\"height:1px; width:100%;background-color:#4178a1; border: 0 none\"\/>\n<hr style=\"height:6px; width:100%;background-color:#4178a1; border: 0 none\"\/>\n<hr style=\"height:1px; width:100%;background-color:#4178a1; border: 0 none\"\/>\n<p style=\"margin:40px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 2<\/strong> <\/p>\n<p><strong> Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]2[\/tex] se, e somente se, terminar em  [tex]0[\/tex], ou [tex]2[\/tex], ou [tex]4[\/tex], ou [tex]6[\/tex], ou [tex]8[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\" style=\"width:90%;\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 15638748 \u00e9 divis\u00edvel por 2, pois termina em 8.<\/li>\n<li> 6749029876539871375986 \u00e9 divis\u00edvel por 2, pois termina em 6.<\/li>\n<li> 7629817 n\u00e3o \u00e9 divis\u00edvel por 2, pois termina em 7.<\/li>\n<li> 2578014971 n\u00e3o \u00e9 divis\u00edvel por 2, pois termina em 1.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 3<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]3[\/tex] se, e somente se, a soma de seus algarismos for divis\u00edvel por [tex]3[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 3741 \u00e9 divis\u00edvel por 3, pois 3+7+4+1=15 e 15 \u00e9 divis\u00edvel por 3.<\/li>\n<li> 100104053 n\u00e3o \u00e9 divis\u00edvel por 3, pois 1+0+0+1+0+4+0+5+3=14 e 14 n\u00e3o \u00e9 divis\u00edvel por 3.<\/li>\n<li> 80200010103 \u00e9 divis\u00edvel por 3, pois 8+0+2+0+0+0+1+0+1+0+3=15 e 15 \u00e9 divis\u00edvel por 3.<\/li>\n<li> 738295698 tamb\u00e9m \u00e9 divis\u00edvel por 3. Observe que<br \/>\n7+3+8+2+9+5+6+9+8=57 &nbsp; &nbsp; e &nbsp;&nbsp; 5+7=12.<br \/>\nComo 12 \u00e9 divis\u00edvel por 3, 57 \u00e9 divis\u00edvel por 3, portanto 738295698 tamb\u00e9m o \u00e9.<\/li>\n<li> 19274654327498765128376538476253849631 tamb\u00e9m \u00e9 divis\u00edvel por 3. Observe que<br \/>\n1+9+2+7+4+6+5+4+3+2+7+4+9+8+7+6+5+1+2+8+3+7+6+5+3+8+4+7<br \/>\n+6+2+5+3+8+4+9+6+3+1=190 &nbsp;&nbsp; e &nbsp;&nbsp; 1+9+0=10.<br \/>\nComo 10 n\u00e3o \u00e9 divis\u00edvel por 3, 190 n\u00e3o \u00e9 divis\u00edvel por 3, portanto 19274654327498765128376538476253849631 tamb\u00e9m n\u00e3o \u00e9.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 4<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex], com mais de dois algarismos, \u00e9 divis\u00edvel por [tex]4[\/tex] se, e somente se, o n\u00famero formado por seus dois \u00faltimos algarismos for divis\u00edvel por [tex]4[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 12898<strong>24<\/strong> \u00e9 divis\u00edvel por 4, pois 24 \u00e9 divis\u00edvel por 4.<\/li>\n<li> 789612874536<strong>16<\/strong> \u00e9 divis\u00edvel por 4, pois 16 \u00e9 divis\u00edvel por 4.<\/li>\n<li> 2487685645<strong>18<\/strong> n\u00e3o \u00e9 divis\u00edvel por 4, pois 18 n\u00e3o \u00e9 divis\u00edvel por 4.<\/li>\n<li> 75123987569024059750125874<strong>34<\/strong> n\u00e3o \u00e9 divis\u00edvel por 4, pois 34 n\u00e3o \u00e9 divis\u00edvel por 4.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 5<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]5[\/tex] se, e somente se, terminar em  [tex]0[\/tex], ou [tex]5[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 2346560 \u00e9 divis\u00edvel por 5, pois termina em 0.<\/li>\n<li> 98387468597839789354905698115 \u00e9 divis\u00edvel por 5, pois termina em 5.<\/li>\n<li> 889977556699881234 n\u00e3o  \u00e9 divis\u00edvel por 5, pois termina em 4.<\/li>\n<li> 18635987356259875410397602397423029737 n\u00e3o  \u00e9 divis\u00edvel por 5, pois termina em 7.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 6<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]6[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel, simultaneamente, por [tex]2[\/tex] e [tex]3[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 618024 \u00e9 divis\u00edvel por 6, pois<br \/>\n618024 \u00e9 divis\u00edvel por 2 (termina em 4);<br \/>\n618024 \u00e9 divis\u00edvel por 3 (6+1+8+0+2+4=21 que \u00e9 divis\u00edvel por 3).<\/li>\n<\/ul>\n<ul>\n<li> 87400012 n\u00e3o \u00e9 divis\u00edvel por 6, pois, embora seja  divis\u00edvel por 2 (termina em 2), 87400012 n\u00e3o \u00e9 divis\u00edvel por 3 (8+7+4+0+0+0+1+2=22 e 22 n\u00e3o \u00e9 divis\u00edvel por 3).<\/li>\n<\/ul>\n<ul>\n<li> 2451093 n\u00e3o \u00e9 divis\u00edvel por 6, pois, embora seja  divis\u00edvel por 3 (2+4+5+1+0+9+3=24  e 24 \u00e9 divis\u00edvel por 3), 2451093 n\u00e3o divis\u00edvel por 2 (termina em 3).<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 7<\/strong> <\/p>\n<p><strong>Um n\u00famero natural <\/strong><span style=\"font-family: Times, 'Times New Roman', serif;font-size:16px;\"><em>n<\/em><\/span> <strong>\u00e9 divis\u00edvel por [tex]7[\/tex] se, e somente se, a diferen\u00e7a entre o<\/strong> <u><strong>n\u00famero obtido de <\/strong><span style=\"font-family: Times, 'Times New Roman', serif;font-size:16px;\"><em>n<\/em><\/span> <strong>retirando-se o algarismo das unidades<\/strong><\/u> <strong>e o<\/strong> <strong><u>dobro do algarismo das unidades<\/u>, tomada <span style=\"color: #6495ED;\">positivamente<\/span>, for divis\u00edvel por [tex]7[\/tex]<\/strong>.<\/p>\n<p style=\"color: #6495ED;font-size:10px; text-align:right;line-height:8px;\">(O positivamente aqui significa que devemos fazer a diferen\u00e7a entre o maior e o menor dos n\u00fameros obtidos.)<\/span><\/p>\n<p>Se o n\u00famero obtido ainda for grande, repete-se o processo, tantas vezes quanto for necess\u00e1rio, at\u00e9 que se possa verificar se o n\u00famero final obtido \u00e9 ou n\u00e3o divis\u00edvel por [tex]7[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\"  text-align: left;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li>[tex]2799588[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]7[\/tex]. Observe:<\/li>\n<\/ul>\n<li>[tex]279958-2\\cdot 8=279958-16=279942[\/tex]<\/li>\n<li>[tex]27994-2\\cdot 2=27994-4=27990[\/tex]<\/li>\n<li>[tex]2799-2\\cdot 0=2799[\/tex]<\/li>\n<li>[tex]279-2\\cdot 9=279-18=261[\/tex]<\/li>\n<li>[tex]26-2\\cdot1=26-2=24[\/tex] e [tex]24[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]7[\/tex]<\/li>\n<ul>\n<li>[tex]5670[\/tex] \u00e9 divis\u00edvel por [tex]7[\/tex]. Observe:<\/li>\n<\/ul>\n<li>[tex]567-2\\cdot 0=567[\/tex]<\/li>\n<li>[tex]56-2\\cdot 7=56-14=42[\/tex]  e [tex]42[\/tex] \u00e9 divis\u00edvel por [tex]7[\/tex]<\/li>\n<ul>\n<li>[tex]100765[\/tex] \u00e9 divis\u00edvel por [tex]7[\/tex]. Observe:<\/li>\n<\/ul>\n<li>[tex]10076-2\\cdot 5=10076-10=10066[\/tex]<\/li>\n<li>[tex]1006-2\\cdot 6=1006-12=994[\/tex]<\/li>\n<li>[tex]99-2\\cdot 4=99-8=91[\/tex]<\/li>\n<li>[tex]9-2\\cdot 1=9-2=7[\/tex]  e [tex]7[\/tex] \u00e9 divis\u00edvel por [tex]7[\/tex]<\/li>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 8<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex], com mais de tr\u00eas algarismos, \u00e9 divis\u00edvel por [tex]8[\/tex] se, e somente se, o n\u00famero formado por seus tr\u00eas \u00faltimos algarismos for divis\u00edvel por [tex]8[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 2<strong>088<\/strong> \u00e9 divis\u00edvel por 8, pois 088=88 \u00e9 divis\u00edvel por 8.<\/li>\n<li> 47680<strong>114<\/strong> n\u00e3o \u00e9 divis\u00edvel por 8, pois 114 n\u00e3o \u00e9 divis\u00edvel por 8.<\/li>\n<li> 892398745632156789<strong>664<\/strong> \u00e9 divis\u00edvel por 8, pois 664 \u00e9 divis\u00edvel por 8.<\/li>\n<li> 859356874523698230002<strong>116<\/strong> n\u00e3o \u00e9 divis\u00edvel por 8, pois 116 n\u00e3o \u00e9 divis\u00edvel por 8.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 9<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]9[\/tex] se, e somente se, a soma de seus algarismos for divis\u00edvel por [tex]9[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 5607801 \u00e9 divis\u00edvel por 9, pois 5+6+0+7+8+0+1=27 e 27 \u00e9 divis\u00edvel por 9.<\/li>\n<li> 310579 n\u00e3o \u00e9 divis\u00edvel por 9, pois 3+1+0+5+7+9=25 e 25 n\u00e3o \u00e9 divis\u00edvel por 9.<\/li>\n<li> 80200010103 n\u00e3o \u00e9 divis\u00edvel por 9, pois 8+0+2+0+0+0+1+0+1+0+3=15 e 15 n\u00e3o \u00e9 divis\u00edvel por 9 (embora 15 seja divis\u00edvel por 3 e, consequentemente, o n\u00famero  80200010103 seja divis\u00edvel por 3).<\/li>\n<li> 347382956988716576985 \u00e9 divis\u00edvel por 9. Observe que<br \/>\n3+4+7+3+8+2+9+5+6+9+8+8+7+1+6+5+7+6+9+8+5=126 &nbsp;&nbsp; e &nbsp;&nbsp; 1+2+6=9.<br \/>\nComo 9 \u00e9 divis\u00edvel por 9, 126 \u00e9 divis\u00edvel por 9, portanto 347382956988716576985 tamb\u00e9m o \u00e9.<\/li>\n<li> 29174654327498765128376538476253849637  n\u00e3o \u00e9 divis\u00edvel por 9. Observe que<br \/>\n2+9+1+7+4+6+5+4+3+2+7+4+9+8+7+6+5+1+2+8+3+7+6+5+3+8+4+<br \/>\n+7+6+2+5+3+8+4+9+6+3+7=196 &nbsp;&nbsp; e &nbsp;&nbsp; 1+9+6=16.<br \/>\nComo 16 n\u00e3o \u00e9 divis\u00edvel por 9, 196 n\u00e3o \u00e9 divis\u00edvel por 9; portanto 29174654327498765128376538476253849637  tamb\u00e9m n\u00e3o \u00e9 divis\u00edvel por 9.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 10<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]10[\/tex] se, e somente se, terminar em [tex]0[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 47128520470 \u00e9 divis\u00edvel por 10, pois termina em 0.<\/li>\n<li> 2587460123698520369750687268469793613678 n\u00e3o  \u00e9 divis\u00edvel por 10, pois termina em 8.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 11<\/strong> <\/p>\n<p style=\"margin:20px;\">\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]11[\/tex] se, e somente se,  a diferen\u00e7a entre a <u>soma dos algarismos de ordem \u00edmpar<\/u> e a <u>soma dos algarismos de ordem par<\/u>, tomada <span style=\"color: #6495ED;\">positivamente<\/span>, for um n\u00famero divis\u00edvel por [tex]11[\/tex].<\/strong><br \/>\nAs ordens \u00edmpares e pares devem ser tomadas da direita para esquerda. <\/p>\n<p style=\"color: #6495ED;font-size:10px; text-align:right;line-height:8px;\">O positivamente aqui significa que devemos fazer a diferen\u00e7a entre a maior e a menor somas.<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 60367593 \u00e9 divis\u00edvel por 11; observe que:\n<p style=\"margin:10px;\">\n<p>[tex]\\begin{cases}<br \/>\nS_i=3+5+6+0=14\\\\<br \/>\n\\\\<br \/>\nS_p=9+7+3+6=25<br \/>\n\\end{cases}\\;\\Longrightarrow \\quad S_p-S_i=25-14=11[\/tex] <\/p>\n<p style=\"margin:20px;\">\n<p>e [tex]11[\/tex] \u00e9 divis\u00edvel por [tex]11[\/tex].<\/p>\n<li> 6268110 n\u00e3o \u00e9 divis\u00edvel por 11; observe que:\n<p style=\"margin:10px;\">\n<p>[tex]\\begin{cases}<br \/>\nS_i=0+1+6+6=13\\\\<br \/>\n\\\\<br \/>\nS_p=1+8+2=11<br \/>\n\\end{cases}\\;\\Longrightarrow \\quad S_i-S_p=13-11=2[\/tex] <\/p>\n<p style=\"margin:20px;\">\n<p>e [tex]2[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]11[\/tex].<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 12<\/strong> <\/p>\n<p style=\"margin:20px;\">\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]12[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel, simultaneamente, por [tex]3[\/tex] e [tex]4[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 236018024 \u00e9 divis\u00edvel por 12, pois<br \/>\n236018016 \u00e9 divis\u00edvel por 4 (16 \u00e9 divis\u00edvel por 4);<br \/>\n236018016 \u00e9 divis\u00edvel por 3 (2+3+6+0+1+8+0+1+6=27, que \u00e9 divis\u00edvel por 3).<\/li>\n<\/ul>\n<ul>\n<li> 2350894312 n\u00e3o \u00e9 divis\u00edvel por 12, pois, embora seja  divis\u00edvel por 4 (12 \u00e9 divis\u00edvel por 4), 2350894312 n\u00e3o \u00e9 divis\u00edvel por 3 (2+3+5+0+8+9+4+3+1+2=37 e 37 n\u00e3o \u00e9 divis\u00edvel por 3).<\/li>\n<\/ul>\n<ul>\n<li> 2451093 n\u00e3o \u00e9 divis\u00edvel por 12, pois, embora seja  divis\u00edvel por 3 (2+4+5+1+0+9+3=24  e 24 \u00e9 divis\u00edvel por 3), 2451093 n\u00e3o divis\u00edvel por 4 (93 \u00e9 divis\u00edvel por 4).<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 13<\/strong> <\/p>\n<p><strong>Um n\u00famero natural <\/strong><span style=\"font-family: Times, 'Times New Roman', serif;font-size:16px;\"><em>n<\/em><\/span> <strong>\u00e9 divis\u00edvel por [tex]13[\/tex] se, e somente se, a soma entre o<\/strong> <u><strong>n\u00famero obtido de <\/strong><span style=\"font-family: Times, 'Times New Roman', serif;font-size:16px;\"><em>n<\/em><\/span> <strong>retirando-se o algarismo das unidades<\/strong><\/u> <strong>e o<\/strong> <strong><u>qu\u00e1druplo do algarismo das unidades<\/u> for divis\u00edvel por [tex]13[\/tex]<\/strong>.<br \/>\nSe o n\u00famero obtido ainda for grande, repete-se o processo, tantas vezes quanto for necess\u00e1rio, at\u00e9 que se possa verificar se o n\u00famero final obtido \u00e9 ou n\u00e3o divis\u00edvel por [tex]13[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\"  text-align: left;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li>[tex]16562[\/tex] \u00e9 divis\u00edvel por [tex]13[\/tex]. Observe:<\/li>\n<\/ul>\n<li>[tex]1656+4\\cdot 2=1656+8=1664[\/tex]<\/li>\n<li>[tex]166+4\\cdot 4=166+16=182[\/tex]<\/li>\n<li>[tex]18+4\\cdot 2=18+8=26[\/tex] e [tex]26[\/tex] \u00e9 divis\u00edvel por [tex]13[\/tex]<\/li>\n<ul>\n<li>[tex]33427[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]13[\/tex]. Observe:<\/li>\n<\/ul>\n<li>[tex]3342+4\\cdot 7=3342+28=3370[\/tex]<\/li>\n<li>[tex]337+4\\cdot 0=337[\/tex]<\/li>\n<li>[tex]33+4\\cdot 7=33+28=61[\/tex]\n<li>[tex]6+4\\cdot 1=6+4=10[\/tex] e [tex]10[\/tex] n\u00e3o \u00e9 divis\u00edvel por [tex]13[\/tex]<\/li>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 14<\/strong> <\/p>\n<p style=\"margin:20px;\">\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]14[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel, simultaneamente, por [tex]2[\/tex] e [tex]7[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 7000 \u00e9 divis\u00edvel por 14, pois<br \/>\n7000 \u00e9 divis\u00edvel por 2 (7000 \u00e9 par);<br \/>\n7000 \u00e9 divis\u00edvel por 7.<\/li>\n<\/ul>\n<ul>\n<li> N\u00fameros \u00edmpares n\u00e3o s\u00e3o divis\u00edveis por 14, pois, particularmente, n\u00e3o s\u00e3o divis\u00edveis por 2.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 15<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]15[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel, simultaneamente, por [tex]3[\/tex] e [tex]5[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 60123825 \u00e9 divis\u00edvel por 15, pois<br \/>\n60123825 \u00e9 divis\u00edvel por 3 (6+0+1+2+3+8+2+5 =27 que \u00e9 divis\u00edvel por 3)<br \/>\n60123825 \u00e9 divis\u00edvel por 5 (termina em 5).<\/li>\n<\/ul>\n<ul>\n<li> 210381 n\u00e3o \u00e9 divis\u00edvel por 15, pois, embora seja  divis\u00edvel por 3 (2+1+0+3+8+1= 15 que \u00e9 divis\u00edvel por 3), 210381 n\u00e3o \u00e9 divis\u00edvel por 5 (termina em 1).<\/li>\n<\/ul>\n<ul>\n<li> 811045 n\u00e3o \u00e9 divis\u00edvel por 15, pois, embora seja  divis\u00edvel por 5 (termina em 5), 891045 n\u00e3o divis\u00edvel por 3 (8+1+1+0+4+5=19 que n\u00e3o \u00e9 divis\u00edvel por 3).<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Divisibilidade por 16<\/strong> <\/p>\n<p><strong>Um n\u00famero natural [tex]n[\/tex], com mais de quatro algarismos, \u00e9 divis\u00edvel por [tex]16[\/tex] se, e somente se, o n\u00famero formado por seus quatro \u00faltimos algarismos for divis\u00edvel por [tex]16[\/tex].<\/strong><\/p>\n<p style=\"margin:20px;\">\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li> 23651<strong>0144<\/strong> \u00e9 divis\u00edvel por 16, pois 0144=144 \u00e9 divis\u00edvel por 16.<\/li>\n<li> 892398745632156789<strong>1325<\/strong> n\u00e3o \u00e9 divis\u00edvel por 16, pois 1325 n\u00e3o \u00e9 divis\u00edvel por 16 (\u00e9 \u00edmpar).<\/li>\n<li> 83029854796321452<strong>0032<\/strong> \u00e9 divis\u00edvel por 16, pois 0032=32 \u00e9 divis\u00edvel por 16.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:30px;\">\n<div class=\"gree cb25\" style=\"width:95%;\">\n<p class=\"ti3 c2\" >Um crit\u00e9rio geral<\/strong> <\/p>\n<p><strong>Sejam [tex]a[\/tex] e [tex]b[\/tex] n\u00fameros naturais tais que [tex]mdc(a,b)=1.[\/tex]<br \/>\nUm n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]a\\cdot b[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel simultaneamente por [tex]\\, a \\,[\/tex] e [tex]\\, b.[\/tex]<br \/>\n<\/strong><\/p>\n<div>\n<input class=\"bo1 c2 cb23\" value=\"Exemplos\" 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 = 'Exemplos';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<p style=\"margin:20px;\">\n<div class=\"gree cb24\">\n<p style=\"margin:10px;\">\n<ul>\n<li>Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]45[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel simultaneamente por [tex]5[\/tex] e [tex]9[\/tex].<\/li>\n<li> Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]66[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel simultaneamente por [tex]6[\/tex] e [tex]11[\/tex].<\/li>\n<li> Um n\u00famero natural [tex]n[\/tex] \u00e9 divis\u00edvel por  [tex]99[\/tex] se, e somente se,  [tex]n[\/tex] for divis\u00edvel simultaneamente por [tex]9[\/tex] e [tex]11[\/tex].<\/li>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:40px;\">\n<hr style=\"height:2px;background-color:#4178a1; border: 0 none\"\/>\n<hr style=\"height:2px;background-color:#4178a1; 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=51902\" target=\"_blank\"><span style=\"color: #6495ED;\"><strong>Voltar para Sala Principal<\/strong><\/span><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Crit\u00e9rios de divisibilidade Nesta Sala, apresentaremos alguns dos tradicionais crit\u00e9rios de divisibilidade. Fixado um n\u00famero natural n\u00e3o nulo [tex]d[\/tex], um crit\u00e9rio de divisibilidade \u00e9 uma condi\u00e7\u00e3o [tex]P[\/tex] necess\u00e1ria e suficiente para que um n\u00famero natural seja divis\u00edvel por [tex]d[\/tex], portanto algo do tipo: Um n\u00famero natural n \u00e9 divis\u00edvel por d se, e somente se &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/um-pouco-sobre-divisibilidade-criterios-de-divisibilidade-sem-justificativas\/\">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-52717","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/52717","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=52717"}],"version-history":[{"count":27,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/52717\/revisions"}],"predecessor-version":[{"id":169329,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/52717\/revisions\/169329"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=52717"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}