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