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HMMT 二月 2004 · 代数 · 第 2 题

HMMT February 2004 — Algebra — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Find the largest number n such that (2004!)! is divisible by (( n !)!)!.
解析
  1. Find the largest number n such that (2004!)! is divisible by (( n !)!)!. Solution: 6 For positive integers a, b , we have a ! | b ! ⇔ a ! ≤ b ! ⇔ a ≤ b. Thus, (( n !)!)! | (2004!)! ⇔ ( n !)! ≤ 2004! ⇔ n ! ≤ 2004 ⇔ n ≤ 6 .