HMMT 二月 2005 · 冲刺赛 · 第 1 题

英文解析

1. Find the largest positive integer n such that 1 + 2 + 3 + · · · + n is divisible by 1 + 2 +
   3 + · · · + n .
   Solution: 1
   The statement is
   2 2
   n ( n + 1) n ( n + 1)
   2 3
   | ⇔ n + 1 | n ( n + 1) = n + n.
   2 2
   2 3
   But n + 1 also divides ( n + 1)( n − n + 2) = n + n + 2, so n + 1 must divide 2. Hence,
   n cannot be greater than 1. And n = 1 clearly works, so that is the answer.