{"id":6374,"date":"2021-05-10T06:00:26","date_gmt":"2021-05-10T04:00:26","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6374"},"modified":"2021-05-17T07:56:13","modified_gmt":"2021-05-17T05:56:13","slug":"suma-de-no-multiplos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/suma-de-no-multiplos\/","title":{"rendered":"Suma de no m\u00faltiplos"},"content":{"rendered":"<p>El enunciado del problema 1 de la <a href=\"https:\/\/bit.ly\/3e0f8e4\">Fase Local de la Olimpiada Matem\u00e1tica Espa\u00f1ola del 2011<\/a> es<\/p>\n<blockquote><p>\n  Dado un entero positivo n, hallar la suma de todos los enteros positivos inferiores a 10n que no son m\u00faltiplos de 2 ni de 5.\n<\/p><\/blockquote>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   suma :: Integer -> Integer\n<\/pre>\n<p>tal que (suma n) es la suma de todos los enteros positivos inferiores a 10n que no son m\u00faltiplos de 2 ni de 5. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   suma 7  ==  980\n   length (show (suma (10^(10^5))))  ==  200002\n   length (show (suma (10^(10^6))))  ==  2000002\n   length (show (suma (10^(10^7))))  ==  20000002\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List ((\\\\))\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsuma :: Integer -> Integer\nsuma n =\n  sum [x | x <- [1..10*n],\n           x `mod` 2 \/= 0,\n           x `mod` 5 \/= 0]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nsuma2 :: Integer -> Integer\nsuma2 n =\n  sum ([1..10*n] \\\\ ([2,4..10*n] ++ [5,10..10*n]))\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\n-- Observando los siguientes c\u00e1lculos\n--    \u03bb> map suma [1..10]\n--    [20,80,180,320,500,720,980,1280,1620,2000]\n--    \u03bb> map (`div` 20) it\n--    [1,4,9,16,25,36,49,64,81,100]\n\nsuma3 :: Integer -> Integer\nsuma3 n = 20*n^2\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_suma :: Integer -> Property\nprop_suma n =\n  n > 0 ==>\n  all (== (suma n))\n      [suma2 n,\n       suma3 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_suma\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> suma (2*10^3)\n--    80000000\n--    (0.05 secs, 6,671,720 bytes)\n--    \u03bb> suma2 (2*10^3)\n--    80000000\n--    (2.63 secs, 7,886,190,192 bytes)\n--    \u03bb> suma3 (2*10^3)\n--    80000000\n--    (0.02 secs, 106,736 bytes)\n--\n--    \u03bb> suma (4*10^5)\n--    3200000000000\n--    (2.31 secs, 1,314,056,144 bytes)\n--    \u03bb> suma3 (4*10^5)\n--    3200000000000\n--    (0.02 secs, 106,808 bytes)\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 del problema 1 de la Fase Local de la Olimpiada Matem\u00e1tica Espa\u00f1ola del 2011 es Dado un entero positivo n, hallar la suma de todos los enteros positivos inferiores a 10n que no son m\u00faltiplos de 2 ni de 5. Definir la funci\u00f3n suma :: Integer -> Integer tal que (suma n) es&#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\/6374"}],"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=6374"}],"version-history":[{"count":6,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6374\/revisions"}],"predecessor-version":[{"id":6470,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6374\/revisions\/6470"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6374"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6374"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}