Propertie of the shift permutation

Question

Let $\sigma\in S_n$ such that $\sigma=(12...n)$, choose $k\in\{1,2,...,n\}$ then if $k|n$ and $r=\frac{n}{k}$ then $$\sigma^k=\tau_1\tau_2...\tau_k$$ where each $\tau_i$ is a $r$-cycle.

Is this true? I kind of saw this fact from playing around with the shift operator and dont know how to prove it.

Answer 1

Imagine $n$ beads strung in a circle. If $k|n$ and you count around in steps of size $k$ you return in $n/k$ steps.

You should be able to guess the generalization for more general $k$. It involves the greatest common divisor of $n$ and $k$.