{"id":5788,"date":"2020-04-14T05:30:57","date_gmt":"2020-04-14T03:30:57","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5788"},"modified":"2020-04-21T07:36:37","modified_gmt":"2020-04-21T05:36:37","slug":"orden-de-divisibilidad","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/orden-de-divisibilidad\/","title":{"rendered":"Orden de divisibilidad"},"content":{"rendered":"<p>El orden de divisibilidad de un n\u00famero x es el mayor n tal que para todo i menor o igual que n, los i primeros d\u00edgitos de n es divisible por i. Por ejemplo, el orden de divisibilidad de 74156 es 3 porque<\/p>\n<pre lang=\"text\">\n   7       es divisible por 1\n   74      es divisible por 2\n   741     es divisible por 3\n   7415 no es divisible por 4\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   ordenDeDivisibilidad :: Integer -> Int\n<\/pre>\n<p>tal que (ordenDeDivisibilidad x) es el orden de divisibilidad de x. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   ordenDeDivisibilidad 74156                      ==  3\n   ordenDeDivisibilidad 12                         ==  2\n   ordenDeDivisibilidad 7                          ==  1\n   ordenDeDivisibilidad 3608528850368400786036725  ==  25\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (inits)\n\n-- 1\u00aa definici\u00f3n de ordenDeDivisibilidad\n-- =====================================\n\nordenDeDivisibilidad :: Integer -> Int\nordenDeDivisibilidad n = \n  length (takeWhile (\\(x,k) -> x `mod` k == 0) (zip (sucDigitos n) [1..]))\n\n-- (sucDigitos x) es la sucesi\u00f3n de los d\u00edgitos de x. Por ejemplo,\n--    sucDigitos 325    ==  [3,32,325]\n--    sucDigitos 32050  ==  [3,32,320,3205,32050]\nsucDigitos :: Integer -> [Integer]\nsucDigitos n = \n    [n `div` (10^i) | i <- [k-1,k-2..0]]\n    where k = length (show n)\n\n-- 2\u00aa definici\u00f3n de sucDigitos\nsucDigitos2 :: Integer -> [Integer]\nsucDigitos2 n = [read xs | xs <- aux (show n)]\n  where aux []     = []\n        aux (d:ds) = [d] : map (d:) (aux ds)\n\n-- 3\u00aa definici\u00f3n de sucDigitos\nsucDigitos3 :: Integer -> [Integer]\nsucDigitos3 n = \n  [read (take k ds) | k <- [1..length ds]]\n  where ds = show n\n\n-- 4\u00aa definici\u00f3n de sucDigitos\nsucDigitos4 :: Integer -> [Integer]\nsucDigitos4 n = [read xs | xs <- tail (inits (show n))]\n\n-- 5\u00aa definici\u00f3n de sucDigitos\nsucDigitos5 :: Integer -> [Integer]\nsucDigitos5 n = map read (tail (inits (show n)))\n\n-- 6\u00aa definici\u00f3n de sucDigitos\nsucDigitos6 :: Integer -> [Integer]\nsucDigitos6 = map read . (tail . inits . show)\n\n-- Eficiencia de las definiciones de sucDigitos\n--    ghci> length (sucDigitos (10^5000))\n--    5001\n--    (0.01 secs, 1550688 bytes)\n--    ghci> length (sucDigitos2 (10^5000))\n--    5001\n--    (1.25 secs, 729411872 bytes)\n--    ghci> length (sucDigitos3 (10^5000))\n--    5001\n--    (0.02 secs, 2265120 bytes)\n--    ghci> length (sucDigitos4 (10^5000))\n--    5001\n--    (1.10 secs, 728366872 bytes)\n--    ghci> length (sucDigitos5 (10^5000))\n--    5001\n--    (1.12 secs, 728393864 bytes)\n--    ghci> length (sucDigitos6 (10^5000))\n--    5001\n--    (1.20 secs, 728403052 bytes)\n-- \n--    ghci> length (sucDigitos (10^3000000))\n--    3000001\n--    (2.73 secs, 820042696 bytes)\n--    ghci> length (sucDigitos3 (10^3000000))\n--    3000001\n--    (3.69 secs, 820043688 bytes)\n\n-- 2\u00aa definici\u00f3n de ordenDeDivisibilidad\n-- =====================================\n\nordenDeDivisibilidad2 :: Integer -> Int\nordenDeDivisibilidad2 x =\n  length\n  $ takeWhile (==0)\n  $ zipWith (mod . read) (tail $ inits $ show x) [1..]\n<\/pre>\n<h4>Otras soluciones<\/h4>\n<ul>\n<li>Se pueden escribir otras soluciones en los comentarios.\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 orden de divisibilidad de un n\u00famero x es el mayor n tal que para todo i menor o igual que n, los i primeros d\u00edgitos de n es divisible por i. Por ejemplo, el orden de divisibilidad de 74156 es 3 porque 7 es divisible por 1 74 es divisible por 2 741 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":[4],"tags":[8,30,74,28,10,89,11,95,6,33,45,47,34,9],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5788"}],"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=5788"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5788\/revisions"}],"predecessor-version":[{"id":5815,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5788\/revisions\/5815"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5788"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5788"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5788"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}