返回题库

HMMT 二月 2006 · TEAM2 赛 · 第 11 题

HMMT February 2006 — TEAM2 Round — Problem 11

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [15] Find the largest positive integer n such that 1! + 2! + 3! + · · · + n ! is a perfect square. Prove thatyour answer is correct.
    2 2 2
解析

英文解析

  1. [15] Find the largest positive integer n such that 1! + 2! + 3! + · · · + n ! is a perfectsquare. Prove that your answer is correct.
    Answer: 33
    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.