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HMMT 二月 2006 · TEAM2 赛 · 第 11 题

HMMT February 2006 — TEAM2 Round — Problem 11

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [15] Find the largest positive integer n such that 1! + 2! + 3! + · · · + n ! is a perfect square. Prove that your answer is correct. 2 2 2
解析
  1. [15] Find the largest positive integer n such that 1! + 2! + 3! + · · · + n ! is a perfect square. Prove that your answer is correct. 3 Answer: 3 Solution: Clearly 1! + 2! + 3! = 9 works. For n ≥ 4, we have 1! + 2! + 3! + · · · + n ! ≡ 1! + 2! + 3! + 4! ≡ 3 (mod 5) , but there are no squares congruent to 3 modulo 5.