HMMT 2017 T9

Let $n$ be a positive odd integer greater than 2 , and consider a regular $n$-gon $\mathcal{G}$ in the plane centered at the origin. Let a subpolygon $\mathcal{G}^{\prime}$ be a polygon with at least 3 vertices whose vertex set is a subset of that of $\mathcal{G}$. Say $\mathcal{G}^{\prime}$ is well-centered if its centroid is the origin. Also, say $\mathcal{G}^{\prime}$ is decomposable if its vertex set can be written as the disjoint union of regular polygons with at least 3 vertices. Show that all well-centered subpolygons are decomposable if and only if $n$ has at most two distinct prime divisors.