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HMMT 二月 2002 · 冲刺赛 · 第 50 题

HMMT February 2002 — Guts Round — Problem 50

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [6] Give the set of all positive integers n such that ϕ ( n ) = 2002 − 1. k
解析

英文解析

  1. Give the set of all positive integers n such that ϕ ( n ) = 2002 − 1.
    Solution: The empty set, Ø . If m is relatively prime to n and m < n , then n − mmust likewise be relatively prime to n , and these are distinct for n > 2 since n/ 2 , n arenot relatively prime. Therefore, for all n > 2, ϕ ( n ) must be even. 2002 − 1 is odd, and 2
    ϕ (2) = 1 6 = 2002 − 1, so no numbers n fulfill the equation.2 k