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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.

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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?

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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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