AMC 8 2026 Problem 24

AMC8 2026 年第 24 题

整除与因数★★★★☆
The notation n! (read "n factorial") is defined as the product of the first nn positive integers. (For example, 3!=1⋅2⋅3=63! = 1\cdot2\cdot3 = 6.) Define the superfactorial of a positive number, denoted by n!^\widehat{n!}, to be the product of the first nn factorials. (For example, 3!^=1!⋅2!⋅3!=12\widehat{3!} = 1!\cdot2!\cdot3! = 12.) How many factors of 7 appear in the prime factorization of 51!^\widehat{51!}, the superfactorial of 51?
记号 n!（读作"n 阶乘"）定义为前 nn 个正整数的乘积（例如 3!=1⋅2⋅3=63! = 1\cdot2\cdot3 = 6）。 定义超阶乘 n!^\widehat{n!} 为前 nn 个阶乘的乘积（例如 3!^=1!⋅2!⋅3!=12\widehat{3!} = 1!\cdot2!\cdot3! = 12）。在 51!^\widehat{51!} 的质因数分解中，因子 7 出现了多少次？
  1. A.147
  2. B.150
  3. C.156
  4. D.168
  5. E.171

正确答案 · Correct Answer

E

解析 · Solution

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