The roots of unity are the solutions to:
$$z^n=1,z\in \mathbb{C},n\in \mathbb{N}\implies(rcis\theta)^n=1\implies r=1,\theta=\frac{2\pi k}{n},k\in\mathbb{Z}$$
A primitive root of unity for $n$ is a root that is not a root for any lower value of n. I want to count the number of elements in the set of solutions to : $$z^n=1, 1<n<10$$
If $n$ is small, it can be done manually. But if $n$ is large the process becomes tedious. Is there a more elegant way to approach this problem.