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A function on the set of positive divisors

Source: 2022 China TST, Test 2, P5

March 29, 2022
number theoryDivisorsbinomial coefficients

Problem Statement

Given a positive integer nn, let DD is the set of positive divisors of nn, and let f:D→Zf: D \to \mathbb{Z} be a function. Prove that the following are equivalent:
(a) For any positive divisor mm of nn, n ∣ ∑d∣mf(d)(n/dm/d). n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. (b) For any positive divisor kk of nn, k ∣ ∑d∣kf(d). k ~\Big|~ \sum_{d|k} f(d).
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