{"id":6249,"date":"2021-04-06T06:00:51","date_gmt":"2021-04-06T04:00:51","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6249"},"modified":"2021-04-13T07:40:17","modified_gmt":"2021-04-13T05:40:17","slug":"raiz-digital","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/raiz-digital\/","title":{"rendered":"Ra\u00edz digital"},"content":{"rendered":"<p>La ra\u00edz digital de un n\u00famero entero positivo n es el d\u00edgito que resulta al sumar sus d\u00edgitos, volviendo a sumar reiteradamente los resultados de esa suma y de las siguientes hasta que la suma sea un n\u00famero de un d\u00edgito, al que se llama la ra\u00edz digital del n\u00famero n y se representa por D(n). Por ejemplo, la ra\u00edz digital del n\u00famero 23451 es 6, porque 2+3+4+5+1 = 15 y sumando los d\u00edgitos del 15 resulta 6.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   raizDigital :: Integer -> Integer\n<\/pre>\n<p>tal que (raizDigital n) es la ra\u00edz digital del entero positivo n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   raizDigital 23451  ==  6\n<\/pre>\n<p>Comprobar con QuickCheck las siguientes propiedades de la ra\u00edz digital:<\/p>\n<ul>\n<li>D(m + n) = D(D(m) + D(n)).<\/li>\n<li>D(mn) = D(D(m)D(n)).<\/li>\n<li>D(m^n) = D(D(m)^n).<\/li>\n<li>D(D(n)) = D(n).<\/li>\n<li>D(n + 9) = D(n).<\/li>\n<li>D(9n) = 9.<\/li>\n<\/ul>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nraizDigital :: Integer -> Integer\nraizDigital n\n  | n < 10    = n\n  | otherwise = raizDigital (sumaDigitos n)\n\n-- (sumaDigitos n) es la suma de los d\u00edgitos de n. Por ejemplo,\n--    sumaDigitos 2021  ==  5\nsumaDigitos :: Integer -> Integer\nsumaDigitos = sum . digitos\n\n-- (digitos n) es la lista de los d\u00edgitos de n. Por ejemplo,\n--    digitos 2021  ==  [2,0,2,1]\ndigitos :: Integer -> [Integer]\ndigitos x = [read [c] | c <- show x]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nraizDigital2 :: Integer -> Integer\nraizDigital2 n =\n  head (dropWhile (>9) (iterate sumaDigitos n))\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nraizDigital3 :: Integer -> Integer\nraizDigital3 n\n  | m == 0    = 9\n  | otherwise = m\n  where m = n `mod` 9\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nraizDigital4 :: Integer -> Integer\nraizDigital4 n = 1 + (n-1) `mod` 9\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_raizDigital_equiv :: Integer -> Property\nprop_raizDigital_equiv n =\n  n > 0 ==>\n  all (== (raizDigital n))\n      [raizDigital2 n,\n       raizDigital3 n,\n       raizDigital4 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_raizDigital_equiv\n--    +++ OK, passed 100 tests.\n\n-- Propiedades\n-- ===========\n\n-- Las propiedades son\nprop_raizDigital :: Integer -> Integer -> Property\nprop_raizDigital m n =\n  m > 0 && n > 0 ==>\n  d(m + n) == d(d(m) + d(n)) &&\n  d(m*n) == d(d(m)*d(n)) &&\n  d(m^n) == d(d(m)^n) &&\n  d(d(n)) == d(n) &&\n  d(n + 9) == d(n) &&\n  d(9*n) == 9\n  where d = raizDigital\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_raizDigital\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\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>La ra\u00edz digital de un n\u00famero entero positivo n es el d\u00edgito que resulta al sumar sus d\u00edgitos, volviendo a sumar reiteradamente los resultados de esa suma y de las siguientes hasta que la suma sea un n\u00famero de un d\u00edgito, al que se llama la ra\u00edz digital del n\u00famero n y se representa por&#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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6249"}],"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=6249"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6249\/revisions"}],"predecessor-version":[{"id":6281,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6249\/revisions\/6281"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6249"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6249"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}