PMO 2021 Problem 3

Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies

• $f(p) = 1$ for all primes $p$ , and

• $f(ab) = af(b) + bf(a)$ for all $a,b \in \mathbb{Q}^+$ .

For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$ ?