Problem

Source: 2022 China TST, Test 2, P5

Tags: number theory, Divisors, binomial coefficients



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 $n$, \[ k ~\Big|~ \sum_{d|k} f(d). \]