{"id":1675,"date":"2015-11-04T07:55:10","date_gmt":"2015-11-04T05:55:10","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=1675"},"modified":"2015-11-11T08:57:39","modified_gmt":"2015-11-11T06:57:39","slug":"numeros-muy-divisibles-por-3","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-muy-divisibles-por-3\/","title":{"rendered":"N\u00fameros muy divisibles por 3"},"content":{"rendered":"<p>Se dice que un n\u00famero n es <strong>muy divisible por 3<\/strong> si es divisible por 3 y sigue siendo divisible por 3 si vamos quitando d\u00edgitos por la derecha. Por ejemplo, 96060 es muy divisible por 3 porque 96060, 9606, 960, 96 y 9 son todos divisibles por 3.<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   muyDivPor3             :: Integer -> Bool\n   numeroMuyDivPor3Cifras :: Integer -> Integer\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(muyDivPor3 n) se verifica si n es muy divisible por 3. Por ejemplo, <\/li>\n<\/ul>\n<pre lang=\"text\">\n     muyDivPor3 96060 == True\n     muyDivPor3 90616 == False\n<\/pre>\n<ul>\n<li>(numeroMuyDivPor3CifrasC k) es la cantidad de n\u00fameros de k cifras muy divisibles por 3. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     numeroMuyDivPor3Cifras 5                    == 768\n     numeroMuyDivPor3Cifras 7                    == 12288\n     numeroMuyDivPor3Cifras (10^6) `rem` (10^6)  == 332032\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength)\n\nmuyDivPor3 :: Integer -> Bool\nmuyDivPor3 n \n    | n < 10    = n `rem` 3 == 0\n    | otherwise = n `rem` 3 == 0 &#038;&#038; muyDivPor3 (n `div` 10)\n\n-- 1\u00aa definici\u00f3n\n-- =============\n\nnumeroMuyDivPor3Cifras :: Integer -> Integer\nnumeroMuyDivPor3Cifras k = \n    genericLength [x | x <- [10^(k-1)..10^k-1], muyDivPor3 x]\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\nnumeroMuyDivPor3Cifras2 :: Integer -> Integer\nnumeroMuyDivPor3Cifras2 k = \n    genericLength [x | x <- [n,n+3..10^k-1], muyDivPor3 x]\n    where n = k*10^(k-1)\n\n-- 3\u00aa definici\u00f3n\n-- =============\n\nnumeroMuyDivPor3Cifras3 :: Integer -> Integer\nnumeroMuyDivPor3Cifras3 k = genericLength (numeroMuyDivPor3Cifras3a k)\n\nnumeroMuyDivPor3Cifras3a :: Integer -> [Integer]\nnumeroMuyDivPor3Cifras3a 1 = [3,6,9] \nnumeroMuyDivPor3Cifras3a k = \n    [10*x+y | x <- numeroMuyDivPor3Cifras3a (k-1),\n              y <- [0,3..9]]\n\n-- 4\u00aa definici\u00f3n\n-- =============\n\nnumeroMuyDivPor3Cifras4 :: Integer -> Integer\nnumeroMuyDivPor3Cifras4 1 = 3\nnumeroMuyDivPor3Cifras4 k = 4 * numeroMuyDivPor3Cifras4 (k-1)\n\n-- 5\u00aa definici\u00f3n\n-- =============\n\nnumeroMuyDivPor3Cifras5 :: Integer -> Integer\nnumeroMuyDivPor3Cifras5 k = 3 * 4^(k-1)\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> numeroMuyDivPor3Cifras 6\n--    3072\n--    (3.47 secs, 534,789,608 bytes)\n--    \u03bb> numeroMuyDivPor3Cifras2 6\n--    2048\n--    (0.88 secs, 107,883,432 bytes)\n--    \u03bb> numeroMuyDivPor3Cifras3 6\n--    3072\n--    (0.01 secs, 0 bytes)\n--    \n--    \u03bb> numeroMuyDivPor3Cifras2 7\n--    0\n--    (2.57 secs, 375,999,336 bytes)\n--    \u03bb> numeroMuyDivPor3Cifras3 7\n--    12288\n--    (0.02 secs, 0 bytes)\n--    \u03bb> numeroMuyDivPor3Cifras4 7\n--    12288\n--    (0.00 secs, 0 bytes)\n--    \u03bb> numeroMuyDivPor3Cifras5 7\n--    12288\n--    (0.01 secs, 0 bytes)\n--\n--    \u03bb> numeroMuyDivPor3Cifras4 (10^5) `rem` 100000\n--    32032\n--    (5.74 secs, 1,408,600,592 bytes)\n--    \u03bb> numeroMuyDivPor3Cifras5 (10^5) `rem` 100000\n--    32032\n--    (0.02 secs, 0 bytes)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Se dice que un n\u00famero n es muy divisible por 3 si es divisible por 3 y sigue siendo divisible por 3 si vamos quitando d\u00edgitos por la derecha. Por ejemplo, 96060 es muy divisible por 3 porque 96060, 9606, 960, 96 y 9 son todos divisibles por 3. Definir las funciones muyDivPor3 :: Integer&#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":[4],"tags":[8,30,6,31],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1675"}],"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=1675"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1675\/revisions"}],"predecessor-version":[{"id":1705,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1675\/revisions\/1705"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=1675"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=1675"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=1675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}