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HMMT 二月 2005 · 冲刺赛 · 第 1 题

HMMT February 2005 — Guts Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [5] Find the largest positive integer n such that 1 + 2 + 3 + · · · + n is divisible by 1 + 2 + 3 + · · · + n .
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.