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?

Let's say $w=z^6$. We know that $w^4=1$, so $w=\pm 1,\pm i$. Each of these four numbers has $6$ distinct sixth roots.
Does that help?
