{"id":40164,"date":"2025-05-28T09:07:39","date_gmt":"2025-05-28T01:07:39","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/40164.html"},"modified":"2025-05-28T09:07:39","modified_gmt":"2025-05-28T01:07:39","slug":"%e8%93%9d%e6%a1%a5%e6%9d%af2025-5-23%e6%af%8f%e6%97%a5%e4%b8%80%e9%a2%98-%e5%84%bf%e7%ab%a5%e6%95%b0","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/40164.html","title":{"rendered":"\u84dd\u6865\u676f2025.5.23\u6bcf\u65e5\u4e00\u9898-\u513f\u7ae5\u6570"},"content":{"rendered":"

\u513f\u7ae5\u6570<\/h3>\n

\u82e5\u4e00\u4e2a\u6b63\u6574\u6570 <\/p>\n

n <\/p>\n

n <\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u6ee1\u8db3 <\/p>\n

n <\/p>\n

61 <\/p>\n

n^{61} <\/p>\n

<\/span><\/span>n<\/span><\/span>61<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6574\u9664 <\/p>\n

2024 <\/p>\n

! <\/p>\n

2024! <\/p>\n

<\/span><\/span>2024<\/span>!<\/span><\/span><\/span><\/span><\/span>，\u5373 <\/p>\n

2024 <\/p>\n

! <\/p>\n

2024! <\/p>\n

<\/span><\/span>2024<\/span>!<\/span><\/span><\/span><\/span><\/span> \u9664\u4ee5 <\/p>\n

n <\/p>\n

61 <\/p>\n

n^{61} <\/p>\n

<\/span><\/span>n<\/span><\/span>61<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u4f59\u6570\u4e3a <\/p>\n

0 <\/p>\n

0 <\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5219\u79f0 <\/p>\n

n <\/p>\n

n <\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u4e3a\u513f\u7ae5\u6570\u3002<\/p>\n

\u73b0\u5728，\u8bf7\u4f60\u8ba1\u7b97\u5728\u533a\u95f4 <\/p>\n

[ <\/p>\n

1 <\/p>\n

, <\/p>\n

+ <\/p>\n

\u221e <\/p>\n

) <\/p>\n

[1, +\\\\infty) <\/p>\n

<\/span><\/span>[<\/span>1<\/span>,<\/span><\/span>+<\/span>\u221e<\/span>)<\/span><\/span><\/span><\/span><\/span> \u5185\u4e00\u5171\u6709\u591a\u5c11\u4e2a\u513f\u7ae5\u6570\u3002<\/p>\n

\u524d\u7f6e\u77e5\u8bc6<\/p>\n

  • \n

    \u52d2\u8ba9\u5fb7\u516c\u5f0f<\/p>\n

    <\/p>\n

    v <\/p>\n

    p <\/p>\n

    ( <\/p>\n

    n <\/p>\n

    ! <\/p>\n

    ) <\/p>\n

    = <\/p>\n

    \u2211 <\/p>\n

    k <\/p>\n

    = <\/p>\n

    1 <\/p>\n

    \u221e <\/p>\n

    \u230a <\/p>\n

    n <\/p>\n

    p <\/p>\n

    k <\/p>\n

    \u230b <\/p>\n

    v_p(n!) = \\\\sum_{k=1}^{\\\\infty} \\\\left\\\\lfloor \\\\frac{n}{p^k} \\\\right\\\\rfloor <\/p>\n

    <\/span><\/span>v<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>n<\/span>!)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u230a<\/span><\/span><\/span><\/span>p<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u230b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u53ef\u4ee5\u7528\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97 <\/p>\n

    n <\/p>\n

    ! <\/p>\n

    n! <\/p>\n

    <\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span> \u7684\u8d28\u56e0\u6570 <\/p>\n

    p <\/p>\n

    p <\/p>\n

    <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span> \u7684\u6b21\u6570\u3002<\/p>\n<\/li>\n

  • \n

    \u6574\u9664\u7684\u5145\u8981\u6761\u4ef6<\/p>\n

    \u8bbe <\/p>\n

    a <\/p>\n

    a <\/p>\n

    <\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u548c <\/p>\n

    b <\/p>\n

    b <\/p>\n

    <\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u6574\u6570，\u4e14\u5b83\u4eec\u7684\u8d28\u56e0\u6570\u5206\u89e3\u4e3a：<\/p>\n

    <\/p>\n

    a <\/p>\n

    = <\/p>\n

    p <\/p>\n

    1 <\/p>\n

    \u03b1 <\/p>\n

    1 <\/p>\n

    p <\/p>\n

    2 <\/p>\n

    \u03b1 <\/p>\n

    2 <\/p>\n

    \u22ef <\/p>\n

    p <\/p>\n

    k <\/p>\n

    \u03b1 <\/p>\n

    k <\/p>\n

    , <\/p>\n

    b <\/p>\n

    = <\/p>\n

    p <\/p>\n

    1 <\/p>\n

    \u03b2 <\/p>\n

    1 <\/p>\n

    p <\/p>\n

    2 <\/p>\n

    \u03b2 <\/p>\n

    2 <\/p>\n

    \u22ef <\/p>\n

    p <\/p>\n

    k <\/p>\n

    \u03b2 <\/p>\n

    k <\/p>\n

    a = p_1^{\\\\alpha_1} p_2^{\\\\alpha_2} \\\\cdots p_k^{\\\\alpha_k}, \\\\quad b = p_1^{\\\\beta_1} p_2^{\\\\beta_2} \\\\cdots p_k^{\\\\beta_k} <\/p>\n

    <\/span><\/span>a<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>2<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>p<\/span><\/span>k<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>b<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>2<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>p<\/span><\/span>k<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    （\u82e5\u67d0\u4e2a\u8d28\u6570 <\/p>\n

    p <\/p>\n

    i <\/p>\n

    p_i <\/p>\n

    <\/span><\/span>p<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0d\u5728 <\/p>\n

    a <\/p>\n

    a <\/p>\n

    <\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u6216 <\/p>\n

    b <\/p>\n

    b <\/p>\n

    <\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u7684\u5206\u89e3\u4e2d，\u5219\u5bf9\u5e94\u7684\u6307\u6570 <\/p>\n

    \u03b1 <\/p>\n

    i <\/p>\n

    \\\\alpha_i <\/p>\n

    <\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6216 <\/p>\n

    \u03b2 <\/p>\n

    i <\/p>\n

    \\\\beta_i <\/p>\n

    <\/span><\/span>\u03b2<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u89c6\u4e3a <\/p>\n

    0 <\/p>\n

    0 <\/p>\n

    <\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002）<\/p>\n

    \u6574\u9664\u7684\u5145\u8981\u6761\u4ef6<\/p>\n

    <\/p>\n

    a <\/p>\n

    \u2223 <\/p>\n

    b <\/p>\n

    a \\\\mid b <\/p>\n

    <\/span><\/span>a<\/span><\/span>\u2223<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u5f53\u4e14\u4ec5\u5f53 <\/p>\n

    \u2200 <\/p>\n

    i <\/p>\n

    \u2208 <\/p>\n

    { <\/p>\n

    1 <\/p>\n

    , <\/p>\n

    2 <\/p>\n

    , <\/p>\n

    \u2026 <\/p>\n

    , <\/p>\n

    k <\/p>\n

    } <\/p>\n

    , <\/p>\n

    \u03b1 <\/p>\n

    i <\/p>\n

    \u2264 <\/p>\n

    \u03b2 <\/p>\n

    i <\/p>\n

    \\\\forall i \\\\in \\\\{1, 2, \\\\dots, k\\\\}, \\\\alpha_i \\\\leq \\\\beta_i <\/p>\n

    <\/span><\/span>\u2200<\/span>i<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>{<\/span>1<\/span>,<\/span><\/span>2<\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>k<\/span>}<\/span>,<\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    \u5373 <\/p>\n

    a <\/p>\n

    a <\/p>\n

    <\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u7684\u6240\u6709\u8d28\u56e0\u6570\u7684\u5e42\u6b21\u90fd\u4e0d\u8d85\u8fc7 <\/p>\n

    b <\/p>\n

    b <\/p>\n

    <\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u7684\u5bf9\u5e94\u5e42\u6b21\u3002<\/p>\n<\/li>\n

    \u6240\u4ee5\u53ea\u8981\u5206\u89e3 <\/p>\n

    2024 <\/p>\n

    ! <\/p>\n

    2024! <\/p>\n

    <\/span><\/span>2024<\/span>!<\/span><\/span><\/span><\/span><\/span>，\u7136\u540e\u8981\u6ee1\u8db3\u6bcf\u4e2a\u8d28\u56e0\u6570\u7684\u5e42\u6b21 <\/p>\n

    p <\/p>\n

    p <\/p>\n

    <\/span><\/span>p<\/span><\/span><\/span><\/span><\/span> \u6ee1\u8db3 <\/p>\n

    61 <\/p>\n

    p <\/p>\n

    \u2264 <\/p>\n

    e <\/p>\n

    61p \\\\leq e <\/p>\n

    <\/span><\/span>61<\/span>p<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span><\/span>\u3002\u6545\u6700\u7ec8\u7b54\u6848\u4e3a <\/p>\n

    e <\/p>\n

    \/ <\/p>\n

    \/ <\/p>\n

    61 <\/p>\n

    + <\/p>\n

    1 <\/p>\n

    e \/\/ 61 + 1 <\/p>\n

    <\/span><\/span>e<\/span>\/\/61<\/span><\/span>+<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

    def<\/span> is_prime<\/span>(<\/span>x)<\/span>:<\/span>
    \n if<\/span> x<=<\/span>1<\/span>:<\/span>
    \n return<\/span> False<\/span>
    \n for<\/span> i in<\/span> range<\/span>(<\/span>2<\/span>,<\/span>x)<\/span>:<\/span>
    \n if<\/span> x%<\/span>i==<\/span>0<\/span>:<\/span>
    \n return<\/span> False<\/span>
    \n return<\/span> True<\/span><\/p>\n

    d=<\/span>{<\/span>}<\/span>
    \nfor<\/span> i in<\/span> range<\/span>(<\/span>2<\/span>,<\/span>2024<\/span>)<\/span>:<\/span>
    \n if<\/span> is_prime(<\/span>i)<\/span>:<\/span>
    \n cnt=<\/span>0<\/span>
    \n j=<\/span>i
    \n while<\/span> j<=<\/span>2024<\/span>:<\/span>
    \n cnt+=<\/span>2024<\/span>\/\/<\/span>j
    \n j*=<\/span>i
    \n d[<\/span>i]<\/span>=<\/span>cnt<\/p>\n

    # print(d)<\/span>
    \nans=<\/span>1<\/span>
    \nfor<\/span> x,<\/span>cnt in<\/span> d.<\/span>items(<\/span>)<\/span>:<\/span>
    \n if<\/span> cnt>=<\/span>61<\/span>:<\/span>
    \n ans*=<\/span>(<\/span>cnt\/\/<\/span>61<\/span>+<\/span>1<\/span>)<\/span>
    \nprint<\/span>(<\/span>ans)<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

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