CN-TST 2021 Problem 10

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$ , $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$ .