{"id":7460,"date":"2022-10-26T06:00:04","date_gmt":"2022-10-26T04:00:04","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7460"},"modified":"2022-12-14T12:24:14","modified_gmt":"2022-12-14T10:24:14","slug":"algoritmo-de-euclides-del-mcd","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/algoritmo-de-euclides-del-mcd\/","title":{"rendered":"Algoritmo de Euclides del mcd"},"content":{"rendered":"<p>Dados dos n\u00fameros naturales, a y b, es posible calcular su m\u00e1ximo com\u00fan divisor mediante el Algoritmo de Euclides. Este algoritmo se puede resumir en la siguiente f\u00f3rmula:<\/p>\n<pre lang=\"text\">\n   mcd(a,b) = a,                   si b = 0\n            = mcd (b, a m\u00f3dulo b), si b > 0\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   mcd :: Integer -> Integer -> Integer\n<\/pre>\n<p>tal que <code>mcd a b<\/code> es el m\u00e1ximo com\u00fan divisor de <code>a<\/code> y <code>b<\/code> calculado mediante el algoritmo de Euclides. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   mcd 30 45  ==  15\n   mcd 45 30  ==  15\n<\/pre>\n<p>Comprobar con QuickCheck que el m\u00e1ximo com\u00fan divisor de dos n\u00fameros <code>a<\/code> y <code>b<\/code> (ambos mayores que 0) es siempre mayor o igual que 1 y adem\u00e1s es menor o igual que el menor de los n\u00fameros <code>a<\/code>  y <code>b<\/code>.<\/p>\n<p><b>Soluciones<\/b><\/p>\n<p>A continuaci\u00f3n se muestran las <a href=\"#haskell\">soluciones en Haskell<\/a> y las <a href=\"#python\">soluciones en Python<\/a>.<\/p>\n<p><a name=\"haskell\"><\/a><br \/>\n<b>Soluciones en Haskell<\/b><\/p>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\nmcd :: Integer -> Integer -> Integer\nmcd a 0 = a\nmcd a b = mcd b (a `mod` b)\n\n-- La propiedad es\nprop_mcd :: Positive Integer -> Positive Integer -> Bool\nprop_mcd (Positive a) (Positive b) =\n  m >= 1 && m <= min a b\n  where m = mcd a b\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_mcd\n--    OK, passed 100 tests.\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Algoritmo_de_Euclides_del_mcd.hs\">GitHub<\/a>.<\/p>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom hypothesis import given\nfrom hypothesis import strategies as st\n\ndef mcd(a: int, b: int) -> int:\n    if b == 0:\n        return a\n    return mcd(b, a % b)\n\n# -- La propiedad es\n@given(st.integers(min_value=1, max_value=1000),\n       st.integers(min_value=1, max_value=1000))\ndef test_mcd(a: int, b: int) -> None:\n    assert 1 <= mcd(a, b) <= min(a, b)\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q algoritmo_de_Euclides_del_mcd.py\n#    1 passed in 0.22s\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium-Python\/blob\/main\/src\/algoritmo_de_Euclides_del_mcd.py\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dados dos n\u00fameros naturales, a y b, es posible calcular su m\u00e1ximo com\u00fan divisor mediante el Algoritmo de Euclides. Este algoritmo se puede resumir en la siguiente f\u00f3rmula: mcd(a,b) = a, si b = 0 = mcd (b, a m\u00f3dulo b), si b > 0 Definir la funci\u00f3n mcd :: Integer -> Integer -> Integer&#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":[581],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7460"}],"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=7460"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7460\/revisions"}],"predecessor-version":[{"id":7662,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7460\/revisions\/7662"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7460"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7460"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}