2020 AMC 12B Problem 23

How many integers n \geq 2 are there such that whenever z_1, z_2, ..., z_n are complex numbers such that

|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0, then the numbers z_1, z_2, ..., z_n are equally spaced on the unit circle in the complex plane?

\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.