how do you do this?

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9

<p>If such an n existed, then z would necessarily be a complex nth root of unity, meaning that there exists some regular polygon of which z is a vertex. One vertex of this polygon would be at the point (1,0), the others spaced uniformly about the unit circle. By the definition of complex nth roots of unity, n is the number of vertices. </p>

<p>The angular displacement between adjacent vertices would be 30 degrees. The number of vertices is then 360/30 = 12.</p>

<p>So n = 12.</p>