Školjka

%V0 The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality $$\displaystyle f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}$$ 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.