A Bijection between Necklaces and Restricted Multisets

We present a proof of Swee Hong Chan's conjecture establishing a bijection between the set of necklaces of length $n$ with at most $q$ colors, and the set of periodic functions $f: \mathbb{Z}_{n}\to {0, 1, ..., q-1}$ whose weighted sum is divisible by $n$, where $q$ and $n$ are coprime positive integers.