{"id":2857,"date":"2017-01-26T06:00:52","date_gmt":"2017-01-26T04:00:52","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2857"},"modified":"2017-02-02T07:07:38","modified_gmt":"2017-02-02T05:07:38","slug":"maxima-potencia-que-divide-al-factorial","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/maxima-potencia-que-divide-al-factorial\/","title":{"rendered":"M\u00e1xima potencia que divide al factorial"},"content":{"rendered":"<p>La m\u00e1xima potencia de 2 que divide al factorial de 5 es 3, ya que 5! = 120, 120 es divisible por 2^3 y no lo es por 2^4.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   maxPotDivFact :: Integer -> Integer -> Integer\n<\/pre>\n<p>tal que (maxPotDivFact p n), para cada primo p, es el mayor k tal que p^k divide al factorial de n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   maxPotDivFact 2 5       ==  3\n   maxPotDivFact 3 6       ==  2\n   maxPotDivFact 2 10      ==  8\n   maxPotDivFact 3 10      ==  4\n   maxPotDivFact 2 (10^2)  ==  97\n   maxPotDivFact 2 (10^3)  ==  994\n   maxPotDivFact 2 (10^4)  ==  9995\n   maxPotDivFact 2 (10^5)  ==  99994\n   maxPotDivFact 2 (10^6)  ==  999993\n   maxPotDivFact 3 (10^5)  ==  49995\n   maxPotDivFact 3 (10^6)  ==  499993\n   maxPotDivFact 7 (10^5)  ==  16662\n   maxPotDivFact 7 (10^6)  ==  166664\n   length (show (maxPotDivFact 2 (10^20000)))  ==  20000\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericIndex, genericTake)\nimport Data.Numbers.Primes (primes)\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n\nmaxPotDivFact :: Integer -> Integer -> Integer\nmaxPotDivFact p n =\n  head [k | k <- [0..], f `mod` (p^k) \/= 0] - 1\n  where f = product [1..n]\n\n-- 1\u00aa definici\u00f3n\nmaxPotDivFact2 :: Integer -> Integer -> Integer\nmaxPotDivFact2 p n\n  | n < p     = 0\n  | otherwise = m + maxPotDivFact2 p m\n  where m = n `div` p  \n  \n-- 3\u00aa definici\u00f3n\nmaxPotDivFact3 :: Integer -> Integer -> Integer\nmaxPotDivFact3 p = sum . takeWhile (> 0) . tail . iterate (`div` p)\n\n-- Comparaci\u00f3n de eficiencia\n--    \u03bb> maxPotDivFact 2 (10^4)\n--    9995\n--    (5.47 secs, 161,040,624 bytes)\n--    \u03bb> maxPotDivFact2 2 (10^4)\n--    9995\n--    (0.01 secs, 0 bytes)\n--    \u03bb> maxPotDivFact2 2 (10^4)\n--    9995\n--    (0.01 secs, 0 bytes)\n--\n--    \u03bb> length (show (maxPotDivFact2 2 (10^20000)))\n--    20000\n--    (1.93 secs, 592,168,544 bytes)\n--    \u03bb> length (show (maxPotDivFact3 2 (10^20000)))\n--    20000\n--    (2.93 secs, 861,791,504 bytes)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>La m\u00e1xima potencia de 2 que divide al factorial de 5 es 3, ya que 5! = 120, 120 es divisible por 2^3 y no lo es por 2^4. Definir la funci\u00f3n maxPotDivFact :: Integer -> Integer -> Integer tal que (maxPotDivFact p n), para cada primo p, es el mayor k tal que p^k&#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":[7],"tags":[8,30,71,50,89,11,157,6,40,45,34],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2857"}],"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=2857"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2857\/revisions"}],"predecessor-version":[{"id":2892,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2857\/revisions\/2892"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2857"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2857"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2857"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}