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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.

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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 $n$.

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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 $k \mid r$.

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