Question 145
Define $f_k(n)$ as the sum of the $k$-th powers of all positive divisors of the positive integer $n$, that is,
$$
f_k(n) := \sum_{m|n, m \gt 0} m^k.
$$
Find all pairs of positive integers $(a, b)$ such that $f_a(n) \mid f_b(n)$ for all positive integers $n$.
$$
f_k(n) := \sum_{m|n, m \gt 0} m^k.
$$
Find all pairs of positive integers $(a, b)$ such that $f_a(n) \mid f_b(n)$ for all positive integers $n$.