Question 19
For a positive integer $n$, let $\varphi(n)$ represent the number of positive integers not exceeding $n$ and relatively prime to $n$. Find all functions $f:\mathbb{N}_+ \to \mathbb{N}_+$ satisfying that for any positive integers $m, n$ with $m \ge n$,
$$f(m\varphi(n^3)) = f(m)\varphi(n^3).$$
$$f(m\varphi(n^3)) = f(m)\varphi(n^3).$$