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HMMT 二月 2007 · 冲刺赛 · 第 13 题

HMMT February 2007 — Guts Round — Problem 13

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 9 ] Determine the largest integer n such that 7 − 1 is divisible by 2 .
    2 2 2 2
解析

英文解析

  1. [ 9 ] Determine the largest integer n such that 7 − 1 is divisible by 2 .
    Answer: 14 . We have
    ( ) ( ) ( )
    2048 2 4 1024
    7 − 1 = (7 − 1)(7 + 1) 7 + 1 7 + 1 · · · 7 + 1 .
    In the expansion, the eleven terms other than 7 + 1 are divisible by 2 exactly once, as can be checkedeasily with modulo 4.
    2 2 2 2