HMMT 二月 2004 · 冲刺赛 · 第 23 题
HMMT February 2004 — Guts Round — Problem 23
题目详情
- [9] Find the largest integer n such that 3 − 1 is divisible by 2 .
解析
- Find the largest integer n such that 3 − 1 is divisible by 2 . Solution: 11 Write 512 256 256 256 128 128 3 − 1 = (3 + 1)(3 − 1) = (3 + 1)(3 + 1)(3 − 1) 256 128 = · · · = (3 + 1)(3 + 1) · · · (3 + 1)(3 − 1) . k k 2 2 Now each factor 3 + 1, k ≥ 1, is divisible by just one factor of 2, since 3 + 1 = k − 1 k − 1 2 2 2 (3 ) +1 ≡ 1 +1 = 2 (mod 4). Thus we get 8 factors of 2 here, and the remaining terms (3 + 1)(3 − 1) = 8 give us 3 more factors of 2, for a total of 11.