{"id":7110,"date":"2022-07-04T06:00:26","date_gmt":"2022-07-04T04:00:26","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7110"},"modified":"2022-07-06T10:49:26","modified_gmt":"2022-07-06T08:49:26","slug":"ceros-finales-del-factorial","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/ceros-finales-del-factorial\/","title":{"rendered":"Ceros finales del factorial"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   cerosDelFactorial :: Integer -> Integer\n<\/pre>\n<p>tal que <code>(cerosDelFactorial n)<\/code> es el n\u00famero de ceros en que termina el factorial de <code>n<\/code>. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   cerosDelFactorial 24                         == 4\n   cerosDelFactorial 25                         == 6\n   length (show (cerosDelFactorial (10^70000))) == 70000\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength)\nimport Test.QuickCheck (Positive (Positive), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\ncerosDelFactorial1 :: Integer -> Integer\ncerosDelFactorial1 n = ceros (factorial n)\n\n-- (factorial n) es el factorial n. Por ejemplo,\n--    factorial 3  ==  6\nfactorial :: Integer -> Integer\nfactorial n = product [1..n]\n\n-- (ceros n) es el n\u00famero de ceros en los que termina el n\u00famero n. Por\n-- ejemplo, \n--    ceros 320000  ==  4\nceros :: Integer -> Integer\nceros n | rem n 10 \/= 0 = 0\n        | otherwise     = 1 + ceros (div n 10)\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\ncerosDelFactorial2 :: Integer -> Integer\ncerosDelFactorial2 = ceros2 . factorial \n\nceros2 :: Integer -> Integer\nceros2 n = genericLength (takeWhile (=='0') (reverse (show n)))\n\n-- 3\u00aa soluci\u00f3n\n-- =============\n\ncerosDelFactorial3 :: Integer -> Integer\ncerosDelFactorial3 n\n  | n < 5     = 0\n  | otherwise = m + cerosDelFactorial3 m\n  where m = n `div` 5\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_cerosDelFactorial :: Positive Integer -> Bool\nprop_cerosDelFactorial (Positive n) =\n  all (== cerosDelFactorial1 n)\n      [cerosDelFactorial2 n,\n       cerosDelFactorial3 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_cerosDelFactorial\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> cerosDelFactorial1 (4*10^4)\n--    9998\n--    (1.93 secs, 2,296,317,904 bytes)\n--    \u03bb> cerosDelFactorial2 (4*10^4)\n--    9998\n--    (1.57 secs, 1,636,242,040 bytes)\n--    \u03bb> cerosDelFactorial3 (4*10^4)\n--    9998\n--    (0.02 secs, 527,584 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Ceros_finales_del_factorial.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n cerosDelFactorial :: Integer -> Integer tal que (cerosDelFactorial n) es el n\u00famero de ceros en que termina el factorial de n. Por ejemplo, cerosDelFactorial 24 == 4 cerosDelFactorial 25 == 6 length (show (cerosDelFactorial (10^70000))) == 70000 Soluciones import Data.List (genericLength) import Test.QuickCheck (Positive (Positive), quickCheck) &#8212; 1\u00aa soluci\u00f3n &#8212; =========== cerosDelFactorial1&#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":[521],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7110"}],"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=7110"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7110\/revisions"}],"predecessor-version":[{"id":7113,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7110\/revisions\/7113"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7110"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7110"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7110"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}