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HMMT 二月 2014 · 冲刺赛 · 第 6 题

HMMT February 2014 — Guts Round — Problem 6

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Find all integers n for which is an integer. 2 n 4
解析
  1. [ 5 ] Find all integers n for which is an integer. 2 n − 4 2 3 2 ( n +2)( n − 2 n +4) n +8 n − 2 n +4 Answer: 0, 1, 3, 4, 6 We have = = for all n 6 = − 2. Then 2 n − 4 ( n +2)( n − 2) n − 2 2 n − 2 n +4 4 4 = n + , which is an integer if and only if is an integer. This happens when n − 2 n − 2 n − 2 n − 2 = − 4 , − 2 , − 1 , 1 , 2 , 4, corresponding to n = − 2 , 0 , 1 , 3 , 4 , 6, but we have n 6 = − 2 so the answers are 0 , 1 , 3 , 4 , 6.