HMMT 二月 2002 · 冲刺赛 · 第 50 题
HMMT February 2002 — Guts Round — Problem 50
题目详情
- [6] Give the set of all positive integers n such that ϕ ( n ) = 2002 − 1. k
解析
- 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 − m must likewise be relatively prime to n , and these are distinct for n > 2 since n/ 2 , n are 2 not 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. k