2017 AMC 12A Problem 17

The $24$ solutions are the $24$th roots of unity, $z=e^{\frac{\pi i k}{12}}$ for $k=0,1,\ldots,23.$ Then $z^6=e^{\frac{\pi i k}{2}}=\cos\dfrac{k\pi}{2}+i\sin\dfrac{k\pi}{2},$ which is real exactly when $\sin\dfrac{k\pi}{2}=0,$ i.e. when $k$ is even. There are $12$ even values of $k$ in the range. Thus, the correct answer is D.