Question 107
Let $\mathbb{Z}^+$ denote the set of positive integers, and $\text{lcm}(n, m)$ the least positive integer that is divisible by both $n$ and $m$.
Find all functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that
$$
\text{lcm}(n \cdot f(n^2), f(m^2)) = n \cdot f(nm), \text{ for all positive integers } n, m
$$
and such that $f(n)$ divides $n$ for all positive integers $n$.
Find all functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that
$$
\text{lcm}(n \cdot f(n^2), f(m^2)) = n \cdot f(nm), \text{ for all positive integers } n, m
$$
and such that $f(n)$ divides $n$ for all positive integers $n$.