Algebra
# Roots of Unity

Regular polygons are placed on the coordinate plane such that they each have a vertex at $(1,0)$, and centroid at the origin. There is one regular $n$-gon for each value of $n$ between $3$ and $50$, inclusive.

How many values of $n$ are there such that the $n$-gon does not share any vertex other than $(1,0)$ with *any* of the other $n$-gons?

Each of the partial sums $\sum\limits_{k=1}^{n}{e^{k\pi i/3}}$ is graphed on the complex plane for $n\in\{1,2,3,4,5,6\}$.

Which of the $6^\text{th}$ roots of unity coincides with one of these partial sums?