primitive n-th roots of unity Show that the primitive n-th roots of unity have the form $e^{2ki\pi/n}$ for $k,n$ coprime for $0\leq k\leq n$.
Since all primitive n-th roots of unity are n-th roots of unity by definition they all have that form, the question is, how to show $k$ and $n$ are coprime. 

More generally, if $G$ is a cyclic group of order $n$ and $g$ is one of its generators, then the order of $g^k$ is $n/(k,n)$. In particular, the generators are $g^k$ with $(k,n)=1$. 
This is easy to prove: $t$ is the order of $g^k$ iff $t$ is the smallest number such that $1=(g^k)^t=g^{kt}$ iff $kt$ is the smallest multiple of $k$ that is also a multiple of $n$ iff $kt=\operatorname{lcm}(k,n) = kn/(k,n)$ iff $t=n/(k,n)$.
