# Roots of unity divisibility.

Suppose $$r | n$$.

Then $$R:= e^{2i \pi k/r}$$ is an $$n$$-th root of unity. Thus, there exists a unique $$l \in \{0, \dots, n-1\}$$ such that $$R = e^{2\pi i l/n}$$. Does it hold that $$l |n$$?

I tried to prove it using euclidean algorithm but got stuck. This seems very elementary.

No. For instance, if $$r=n$$ then you are asserting that every single element of $$\{0,\dots,n-1\}$$ divides $$n$$, which is obviously false. In general, you can write $$e^{2\pi i l/n}$$ as $$e^{2\pi i k/r}$$ (for an integer $$k$$) iff $$l$$ is divisible by $$n/r$$, in which case you can let $$k=rl/n$$. This certainly doesn't mean that $$l$$ must divide $$n$$.
If $$r \mid n$$, then there exists an integer $$d$$ such that $$dr = n$$. We then have $$k/r = dk/n$$, so the $$l$$ in question should be $$l=dk$$.
Therefore, $$l \mid n$$ if and only if $$k \mid r$$.