Question 186
Given a positive integer $n$, let $D$ be 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 $n$,
$$
n \mid \sum_{d|m} f(d) \binom{n/d}{m/d};
$$
(B) for any positive divisor $k$ of $n$,
$$
k \mid \sum_{d|k} f(d).
$$
(A) for any positive divisor $m$ of $n$,
$$
n \mid \sum_{d|m} f(d) \binom{n/d}{m/d};
$$
(B) for any positive divisor $k$ of $n$,
$$
k \mid \sum_{d|k} f(d).
$$