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 $f:{\mathbb{Z}}_n\to 0,1,\mathrm{.}\mathrm{.}\mathrm{.},q-1$ whose weighted sum is divisible by $n$, where $q$ and $n$ are coprime positive integers.