PUMaC 2010 · 数论（A 组） · 第 4 题
PUMaC 2010 — Number Theory (Division A) — Problem 4
题目详情
- Find the largest positive integer n such that nϕ ( n ) is a perfect square. ( ϕ ( n ) is the number of integers k , 1 ≤ k ≤ n that are relatively prime to n )
解析
- Find the largest positive integer n such that nϕ ( n ) is a perfect square. ( ϕ ( n ) is the number of integers k , 1 ≤ k ≤ n that are relatively prime to n ) Solution: If n > 1, then the highest prime that divides n can be shown to divide nϕ ( n ) to an odd power, and so nϕ ( n ) cannot be a perfect square. It is easy to see that 1 is a perfect square.