A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

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?

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