Q:

How many solutions to $z^{6!}-z^{5!}\in \mathbb{R}$ and $|z|=1$

How many solutions $\Im(z^{720}-z^{120})=0$  where $|z|=1$?  (AIME I $2018$ #$6$)

The following is my interpretation of the first solution.
The next step is to see how many solutions to $\sin(720\theta)-\sin(120\theta)=0$. (1)
One way to do this is consider $\sin(6\omega)-\sin(\omega)$, which has $12$ solutions. (2)
For each solution to (2), corresponds $120$ solutions to (1). I understand that.
But how do I know that $sin(6\omega)=sin(\omega)$ has $12$ solutions?
I'd say during a real contest I could come up with everything but the $12$ solutions part. I tried sketching both graphs on the same plane for $0<x<2\pi$ and I only got 9 intersections, while I can see all $12$ clearly on Desmos, so my drawing is unreliable.  

A:

$$\sin6w=\sin w\iff 6w=2k\pi+w \text{ or } 6w=2k\pi+\pi-w\iff w=\dfrac{2k\pi}{5} \text{ or } w=\dfrac{2k+1}{7}\pi$$
with $k\in\mathbb{Z}$. then
$$w=0, \dfrac{2\pi}{5}, \dfrac{4\pi}{5}, \dfrac{6\pi}{5}, \dfrac{8\pi}{5}, \dfrac{\pi}{7}, \dfrac{3\pi}{7}, \dfrac{5\pi}{7}, \dfrac{7\pi}{7}, \dfrac{9\pi}{7}, \dfrac{11\pi}{7}, \dfrac{13\pi}{7}$$

