let $n = pq$ (prime factorization of n)
let the set $X_n = {\{\omega,\omega^2, \cdots, \omega^n\}}$ the set of $nth$ roots of unity. It can simply be deduced that the sets $X_p$ and $X_q$ are subsets of $X_n$
Conjecture: The set of $kth$ roots of unity with $k\leq n$ denoted by $X_k$ is a subset of $X_n$ if and only if both $k$ and $n-k$ can be expressed as linear combinations of prime factors of $n$