2004 IMO Shortlist S09

The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality $$f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.$$
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$ .

b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.

c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.

从正整数集合 $\mathbb{N}$ 到自身的函数 $f$ 由等式定义 $$f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}。$$

a) 证明对于每两个相对素数 ${m,n\in\mathbb{N}}$ 有 $f(mn)=f(m)f(n)$ 。

b) 证明对于每个$a\in\mathbb{N}$，方程$f(x)=ax$都有解。

c) 找到所有${a\in\mathbb{N}}$，使得方程$f(x)=ax$有唯一解。