2017 AMC 12A Problem 17

There are 24 different complex numbers z such that z^{24}=1. For how many of these is z^6 a real number?

\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 24

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