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HMMT 二月 2011 · TEAM2 赛 · 第 7 题

HMMT February 2011 — TEAM2 Round — Problem 7

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 10 ] Let n be an even positive integer. Prove that ϕ ( n ) ≤ . 2
解析

英文解析

  1. [ 10 ] Let n be an even positive integer. Prove that ϕ ( n ) ≤ .
    Solution: Again, let A be the set of all positive integers x ≤ n such that gcd( n, x ) = 1. Since n is 2
    even, no element of A may be even, and, by definition, every element of A must be at most n . Itnn nfollows that ϕ ( n ), the number of elements of A , must be at most , as desired.nn 2