INMO 2018 Problem 6

Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that $\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$ ; $\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$ .

Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$ .