HMMT 二月 2002 · 冲刺赛 · 第 49 题

HMMT February 2002 — Guts Round — Problem 49

[7] Two integers are relatively prime if they don’t share any common factors, i.e. if their greatest common divisor is 1. Define ϕ ( n ) as the number of positive integers that are less than n and relatively prime to n . Define ϕ ( n ) as the number of positive integers that d are less than dn and relatively prime to n . What is the least n such that ϕ ( n ) = 64000, where x = ϕ ( n ), where y = ϕ ( n )? x y 2

解析

Two integers are relatively prime if they don’t share any common factors, i.e. if their greatest common divisor is 1. Define ϕ ( n ) as the number of positive integers that are less than n and relatively prime to n . Define ϕ ( n ) as the number of positive integers that are d less than dn and relatively prime to n . What is the least n such that ϕ ( n ) = 64000, where x = ϕ ( n ), where y = ϕ ( n )? x y Solution: For fixed n , the pattern of integers relatively prime to n repeats every n 3 integers, so ϕ ( n ) = dϕ ( n ). Therefore the expression in the problem equals ϕ ( n ) . The cube d root of 64000 is 40. ϕ ( p ) = p − 1 for any prime p . Since 40 is one less than a prime, the least n such that ϕ ( n ) = 40 is 41 . 2