ISL 2004-N2

[sourav das]

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.