{"id":8479,"date":"2024-01-29T06:00:50","date_gmt":"2024-01-29T04:00:50","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=8479"},"modified":"2024-02-23T10:44:40","modified_gmt":"2024-02-23T08:44:40","slug":"29-ene-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/29-ene-24\/","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<p><!--more--><\/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\">\nmodule Ceros_finales_del_factorial where\n\nimport Data.List (genericLength)\nimport Test.QuickCheck (Positive (Positive), quickCheck)\nimport Test.Hspec (Spec, describe, hspec, it, shouldBe)\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-- Verificaci\u00f3n\n-- ============\n\nverifica :: IO ()\nverifica = hspec spec\n\nspecG :: (Integer -> Integer) -> Spec\nspecG cerosDelFactorial = do\n  it \"e1\" $\n    cerosDelFactorial 24 `shouldBe` 4\n  it \"e2\" $\n    cerosDelFactorial 25 `shouldBe` 6\n\nspec :: Spec\nspec = do\n  describe \"def. 1\" $ specG cerosDelFactorial1\n  describe \"def. 2\" $ specG cerosDelFactorial2\n  describe \"def. 3\" $ specG cerosDelFactorial3\n\n-- La verificaci\u00f3n es\n--    \u03bb> verifica\n--\n--    6 examples, 0 failures\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><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom itertools import takewhile\nfrom math import factorial\nfrom sys import set_int_max_str_digits\nfrom timeit import Timer, default_timer\n\nfrom hypothesis import given\nfrom hypothesis import strategies as st\n\nset_int_max_str_digits(10**6)\n\n# 1\u00aa soluci\u00f3n\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\ndef ceros(n: int) -> int:\n    r = 0\n    while n % 10 == 0 and n != 0:\n        r += 1\n        n \/\/= 10\n    return r\n\ndef cerosDelFactorial1(n: int) -> int:\n    return ceros(factorial(n))\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef ceros2(n: int) -> int:\n    return len(list(takewhile(lambda x: x == '0', reversed(str(n)))))\n\ndef cerosDelFactorial2(n: int) -> int:\n    return ceros2(factorial(n))\n\n# 3\u00aa soluci\u00f3n\n# =============\n\ndef cerosDelFactorial3(n: int) -> int:\n    r = 0\n    while n >= 5:\n        n = n \/\/ 5\n        r += n\n    return r\n\n# Verificaci\u00f3n\n# ============\n\ndef test_cerosDelFactorial() -> None:\n    for cerosDelFactorial in [cerosDelFactorial1,\n                              cerosDelFactorial2,\n                              cerosDelFactorial3]:\n        assert cerosDelFactorial(24) == 4\n        assert cerosDelFactorial(25) == 6\n    print(\"Verificado\")\n\n# La verificaci\u00f3n es\n#    >>> test_cerosDelFactorial()\n#    Verificado\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\n@given(st.integers(min_value=0, max_value=1000))\ndef test_cerosDelFactorial_equiv(n: int) -> None:\n    r = cerosDelFactorial1(n)\n    assert cerosDelFactorial2(n) == r\n    assert cerosDelFactorial3(n) == r\n\n# La comprobaci\u00f3n es\n#    >>> test_cerosDelFactorial_equiv()\n#    >>>\n\n# Comparaci\u00f3n de eficiencia\n# =========================\n\ndef tiempo(e: str) -> None:\n    \"\"\"Tiempo (en segundos) de evaluar la expresi\u00f3n e.\"\"\"\n    t = Timer(e, \"\", default_timer, globals()).timeit(1)\n    print(f\"{t:0.2f} segundos\")\n\n# La comparaci\u00f3n es\n#    >>> tiempo('cerosDelFactorial1(6*10**4)')\n#    2.47 segundos\n#    >>> tiempo('cerosDelFactorial2(6*10**4)')\n#    0.77 segundos\n#    >>> tiempo('cerosDelFactorial3(6*10**4)')\n#    0.00 segundos\n\n# Comprobaci\u00f3n de todas las propiedades\n# =====================================\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -v Ceros_finales_del_factorial.py\n#       test_cerosDelFactorial PASSED\n#       test_cerosDelFactorial_equiv PASSED\n<\/pre>\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<\/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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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\/8479"}],"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=8479"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8479\/revisions"}],"predecessor-version":[{"id":8500,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8479\/revisions\/8500"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8479"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8479"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8479"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}