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

This is one of the AMC problems from this year. I've been trying to solve it, but I couldn't and a peek at the answers (not recommended, I know) talked about Euler's theorem etc., which I haven't learnt yet. Is there an 'easier' way to solve this problem?

The $n$th complex roots of $1$ are the points on the complex unit circle with angles of $\frac{2\pi k}n, k\in \Bbb N$. This looks something like this (for $n=7$):

Now imagine this picture with $n=24$, and $n=6$. Where do they overlap?
