{"id":5102,"date":"2019-11-12T06:00:55","date_gmt":"2019-11-12T04:00:55","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5102"},"modified":"2019-11-20T08:54:35","modified_gmt":"2019-11-20T06:54:35","slug":"ultimo-digito-no-nulo-del-factorial","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/ultimo-digito-no-nulo-del-factorial\/","title":{"rendered":"\u00daltimo d\u00edgito no nulo del factorial"},"content":{"rendered":"<p>El factorial de 7 es<\/p>\n<pre lang=\"text\">\n   7! = 1 * 2 * 3 * 4 * 5 * 6 * 7 = 5040\n<\/pre>\n<p>por tanto, el \u00faltimo d\u00edgito no nulo del factorial de 7 es 4.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   ultimoNoNuloFactorial :: Integer -> Integer\n<\/pre>\n<p>tal que (ultimoNoNuloFactorial n) es el \u00faltimo d\u00edgito no nulo del factorial de n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   ultimoNoNuloFactorial  7  == 4\n   ultimoNoNuloFactorial 10  == 8\n   ultimoNoNuloFactorial 12  == 6\n   ultimoNoNuloFactorial 97  == 2\n   ultimoNoNuloFactorial  0  == 1\n<\/pre>\n<p>Comprobar con QuickCheck que si n es mayor que 4, entonces el \u00faltimo d\u00edgito no nulo del factorial de n es par.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Char (digitToInt)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nultimoNoNuloFactorial :: Integer -> Integer\nultimoNoNuloFactorial = ultimoNoNulo . factorial\n\n-- (factorial n) es el factorial de n. Por ejemplo,\n--    factorial 7  ==  5040\nfactorial :: Integer -> Integer\nfactorial n = product [1..n]\n\n-- (ultimoNoNulo n) es el \u00faltimo d\u00edgito no nulo de n. Por ejemplo,\n--    ultimoNoNulo 5040  ==  4\nultimoNoNulo :: Integer -> Integer\nultimoNoNulo n | r \/= 0    = r\n               | otherwise = ultimoNoNulo q\n  where (q,r) = n `quotRem` 10\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nultimoNoNuloFactorial2 :: Integer -> Integer\nultimoNoNuloFactorial2 = last . filter (\/= 0) . digitos . factorial\n\ndigitos :: Integer -> [Integer]\ndigitos n = [read [x] | x <- show n]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nultimoNoNuloFactorial3 :: Integer -> Integer\nultimoNoNuloFactorial3 = last . filter (\/= 0) . digitos3 . factorial3\n\ndigitos3 :: Integer -> [Integer]\ndigitos3 = map (fromIntegral . digitToInt) . show\n\nfactorial3 :: Integer -> Integer\nfactorial3 = product . enumFromTo 1\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nultimoNoNulo4 :: Integer -> Integer\nultimoNoNulo4 n = read [head (dropWhile (=='0') (reverse (show n)))]\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\nultimoNoNulo5 :: Integer -> Integer\nultimoNoNulo5 =\n  read . return . head . dropWhile ('0' ==) . reverse . show\n\n-- Propiedad\n-- =========\n\n-- La propiedad es\nprop_ultimoNoNuloFactorial :: Integer -> Property\nprop_ultimoNoNuloFactorial n = \n  n > 4 ==> even (ultimoNoNuloFactorial n)\n                  \n-- La comprobaci\u00f3n es\n--    ghci> quickCheck prop_ultimoNoNuloFactorial\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nBusca el tu esencial,<br \/>\nque no est\u00e1 en ninguna parte<br \/>\ny en todas partes est\u00e1.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>El factorial de 7 es 7! = 1 * 2 * 3 * 4 * 5 * 6 * 7 = 5040 por tanto, el \u00faltimo d\u00edgito no nulo del factorial de 7 es 4. Definir la funci\u00f3n ultimoNoNuloFactorial :: Integer -> Integer tal que (ultimoNoNuloFactorial n) es el \u00faltimo d\u00edgito no nulo del factorial&#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":[5],"tags":[8,30,59,91,71,11,157,6,31,32,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5102"}],"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=5102"}],"version-history":[{"count":6,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5102\/revisions"}],"predecessor-version":[{"id":5146,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5102\/revisions\/5146"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5102"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5102"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}