HMMT 二月 2004 · 冲刺赛 · 第 13 题
HMMT February 2004 — Guts Round — Problem 13
题目详情
- [7] Find all positive integer solutions ( m, n ) to the following equation: 2 m = 1! + 2! + · · · + n ! . a n +2
解析
- Find all positive integer solutions ( m, n ) to the following equation: 2 m = 1! + 2! + · · · + n ! . 3 Solution: (1 , 1) , (3 , 3) A square must end in the digit 0 , 1 , 4 , 5 , 6, or 9. If n ≥ 4, then 1! + 2! + · · · + n ! ends in the digit 3, so cannot be a square. A simple check for the remaining cases reveals that the only solutions are (1 , 1) and (3 , 3). a n +2