{"id":6475,"date":"2021-05-28T06:00:04","date_gmt":"2021-05-28T04:00:04","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6475"},"modified":"2021-06-04T07:55:07","modified_gmt":"2021-06-04T05:55:07","slug":"multiplos-sin-ceros","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/multiplos-sin-ceros\/","title":{"rendered":"M\u00faltiplos sin ceros"},"content":{"rendered":"<p>El enunciado de un problema para la IMO (Olimpiada Internacional de Matem\u00e1ticas) de 1972 es<\/p>\n<blockquote><p>\n  Demostrar que cada n \u2262 0 (mod 10) posee alg\u00fan m\u00faltiplo sin el d\u00edgito 0.\n<\/p><\/blockquote>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   multiplosSinCeros :: Integer -> [Integer]\n<\/pre>\n<p>tal que (multiplosSinCeros n) es la lista de los m\u00faltiplos de n sin el d\u00edgito 0. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> take 10 (multiplosSinCeros 101)\n   [1111,1212,1313,1414,1515,1616,1717,1818,1919,2121]\n<\/pre>\n<p>Comprobar con QuickCheck que si n es un n\u00famero entero positivo no divisible por 10, entonces n posee alg\u00fan m\u00faltiplo sin el d\u00edgito 0.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck (Property(..), (==>), quickCheck)\n\nmultiplosSinCeros :: Integer -> [Integer]\nmultiplosSinCeros n =\n  filter sinCeros [n,2*n..]\n\nsinCeros :: Integer -> Bool\nsinCeros n =\n  '0' `notElem` show n\n\n-- La propiedad es\nprop_multiplosSinCeros :: Integer -> Property\nprop_multiplosSinCeros n =\n  n > 0 && n `mod` 10 \/= 0 ==>\n  not (null (multiplosSinCeros n))\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_multiplosSinCeros\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>El enunciado de un problema para la IMO (Olimpiada Internacional de Matem\u00e1ticas) de 1972 es Demostrar que cada n \u2262 0 (mod 10) posee alg\u00fan m\u00faltiplo sin el d\u00edgito 0. Definir la funci\u00f3n multiplosSinCeros :: Integer -> [Integer] tal que (multiplosSinCeros n) es la lista de los m\u00faltiplos de n sin el d\u00edgito 0. Por&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[2],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6475"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=6475"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6475\/revisions"}],"predecessor-version":[{"id":6510,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6475\/revisions\/6510"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6475"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6475"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6475"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}