CIIM 2019 Problem 6

Determine all the injective functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, such that for each pair of integers $(m, n)$ the following conditions hold:

$a)$ $f(mn) = f(m)f(n)$ $b)$ $f(m^2 + n^2) \mid f(m^2) + f(n^2).$