{"id":2519,"date":"2022-06-27T20:41:40","date_gmt":"2022-06-27T20:41:40","guid":{"rendered":"https:\/\/blogs.ua.es\/dimates\/?p=2519"},"modified":"2022-06-27T20:41:40","modified_gmt":"2022-06-27T20:41:40","slug":"solucion-a-numero-de-divisores-2","status":"publish","type":"post","link":"https:\/\/blogs.ua.es\/dimates\/2022\/06\/27\/solucion-a-numero-de-divisores-2\/","title":{"rendered":"Soluci\u00f3n a n\u00famero de divisores"},"content":{"rendered":"<pre>Problema 1 del concurso Marat\u00f3 de problemes 2022\r\nSe dirige a una edad de: 14-15 a\u00f1os<\/pre>\n<p>El n\u00famero 20 = 2\u00b2\u00b75 tiene 6 divisores enteros positivos, el n\u00famero 22 = 2\u00b711 tiene 4 y el 2022 = 2\u00b73\u00b7337 tiene 8.<\/p>\n<p>\u00bfCu\u00e1ntos divisores enteros positivos tiene 20\u00b2\u00b2?<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2516\" src=\"https:\/\/blogs.ua.es\/dimates\/files\/2022\/06\/251.divisores.png\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/blogs.ua.es\/dimates\/files\/2022\/06\/251.divisores.png 300w, https:\/\/blogs.ua.es\/dimates\/files\/2022\/06\/251.divisores-150x150.png 150w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\nSoluci\u00f3n:<br \/>\n<!--more--><br \/>\nUna de las estrategias, si no conoces la f\u00f3rmula, es descomponer los divisores en factores.<\/p>\n<p>En el caso de 20, que tiene 6 divisores, ser\u00edan los siguientes:<\/p>\n<p>Los que s\u00f3lo tienen factores 2: 1, 2, 2\u00b2.<\/p>\n<p>Los que tienen un \u00fanico factor 5: 5, 2\u00b75, 2\u00b2\u00b75.<\/p>\n<p>Otro ejemplo, para el 2022 = 2\u00b73\u00b7337.<\/p>\n<p>Tendr\u00edamos los divisores que s\u00f3lo tienen el factor 2: 1, 2.<\/p>\n<p>Los que tienen el factor 3 y el 2: 3, 2\u00b73.<\/p>\n<p>Luego, estos 4 los combinamos con el factor 337:<\/p>\n<p>337, 2\u00b7337;<br \/>\n3\u00b7337, 2\u00b73\u00b7337.<\/p>\n<p>En total los 8 divisores de 2022 (1, 2, 3, 6, 337, 674, 1011 y 2022). Con este sistema no siempre salen ordenados.<\/p>\n<p>De la misma forma, para 20\u00b2\u00b2, tenemos que su descomposici\u00f3n ser\u00eda 2\u2074\u2074\u00b75\u00b2\u00b2, as\u00ed que tendr\u00edamos:<\/p>\n<p>45 divisores que no tienen ning\u00fan 5: 1, 2, 2\u00b2, 2\u00b3, \u2026 , 2\u2074\u2074.<\/p>\n<p>Otros 45 divisores que tienen un \u00fanico 5: 5, 2\u00b75, 2\u00b2\u00b75, 2\u00b3\u00b75, \u2026 , 2\u2074\u2074\u00b75.<\/p>\n<p>Otros 45 m\u00e1s que tienen dos factores 5: 5\u00b2, 2\u00b75\u00b2, 2\u00b2\u00b75\u00b2, 2\u00b3\u00b75\u00b2, \u2026 , 2\u2074\u2074\u00b75\u00b2.<\/p>\n<p>Y as\u00ed sucesivamente, hasta los 45 divisores que tienen un 5\u00b2\u00b2: 5\u00b2\u00b2, 2\u00b75\u00b2\u00b2, 2\u00b2\u00b75\u00b2\u00b2, 2\u00b3\u00b75\u00b2\u00b2, \u2026 , 2\u2074\u2074\u00b75\u00b2\u00b2.<\/p>\n<p>En total ser\u00edan 45\u00b723 = 1035 divisores.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Problema 1 del concurso Marat\u00f3 de problemes 2022 Se dirige a una edad de: 14-15 a\u00f1os El n\u00famero 20 = 2\u00b2\u00b75 tiene 6 divisores enteros positivos, el n\u00famero 22 = 2\u00b711 tiene 4 y el 2022 = 2\u00b73\u00b7337 tiene 8. \u00bfCu\u00e1ntos divisores enteros positivos tiene 20\u00b2\u00b2? Soluci\u00f3n:<\/p>\n","protected":false},"author":4267,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2242026,1738,2849,3303],"tags":[],"class_list":["post-2519","post","type-post","status-publish","format-standard","hentry","category-marato-de-problemes","category-olimpiadas","category-problemas","category-soluciones"],"_links":{"self":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/2519","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/users\/4267"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/comments?post=2519"}],"version-history":[{"count":1,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/2519\/revisions"}],"predecessor-version":[{"id":2520,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/posts\/2519\/revisions\/2520"}],"wp:attachment":[{"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/media?parent=2519"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/categories?post=2519"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ua.es\/dimates\/wp-json\/wp\/v2\/tags?post=2519"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}