CN-TST 2022 Problem 11

Given a positive integer $n$ , let $D$ is the set of positive divisors of $n$ , and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent:

(a) For any positive divisor $m$ of $n$ , $$n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}.$$ (b) For any positive divisor $k$ of $n$ , $$k ~\Big|~ \sum_{d|k} f(d).$$