返回题库

HMMT 二月 2004 · 代数 · 第 2 题

HMMT February 2004 — Algebra — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Find the largest number n such that (2004!)! is divisible by (( n !)!)!.
解析

英文解析

  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 .