How many complex numbers $z$ are there such that $|z|=1$ and $z^{5040} - z^{720}$ is a real number? My attempt:
If $z^{5040} - z^{720}$ is real, that means that their imaginary parts are equal.
$e^{5040i\theta} - e^{720i\theta} = k$ , where $k$ is a real number
$\sin{5040\theta} = \sin{720\theta} $ 
Let $u=720\theta$
$\sin 7u = \sin u$
By graphing $y=\sin 7u$ and $y=\sin u$, there are $16$ intersections between $0$ and $2\pi$ (including $0$ but not including $2\pi$)
Because the substitution $u=720\theta$ scaled the functions by a factor of $720$ parallel to the x-axis,
there are $11520$ intersections for $\sin{5040\theta} = \sin{720\theta}$ from $\theta=0$ to $\theta= 2\pi$ (not including $2\pi$ because $0+2k\pi=2\pi$ when $k=1$.  
Is $11520$ complex numbers correct? There was no answer provided.

Solve $\sin{7u} = \sin{u} $  as follows,
$$ \sin{7u} - \sin{u} = 2\cos 4u \sin 3u=0$$
$\sin4u = 0$ yields $ u = \frac {\pi n}4$ and $\cos3u=0$ yields $u=\frac\pi6+  \frac {\pi n}3$. Then, 
$$z= e^{ \frac { i2\pi n}{8\cdot 720}}, \>\>\>\>\>
z= e^{ \frac{i\pi}{6\cdot 720 }+i\frac { i2\pi n}{6\cdot 720}}$$
Thus, number of distinct solutions are $8\cdot 720 + 6\cdot 720= 10080$.
