HMMT 二月 2011 · TEAM2 赛 · 第 7 题
HMMT February 2011 — TEAM2 Round — Problem 7
题目详情
- [ 10 ] Let n be an even positive integer. Prove that ϕ ( n ) ≤ . 2
解析
- [ 10 ] Let n be an even positive integer. Prove that ϕ ( n ) ≤ . 2 Solution: Again, let A be the set of all positive integers x ≤ n such that gcd( n, x ) = 1. Since n is n even, no element of A may be even, and, by definition, every element of A must be at most n . It n n n follows that ϕ ( n ), the number of elements of A , must be at most , as desired. n 2