The question asks: if $\alpha$ is an $n$-cycle, then $\alpha^k$ is a product of $gcd(n, k)$ disjoint cycles, each of length $n/gcd(n, k)$.
I am not quite sure how to proceed with the problem, because we know that $\alpha^k(a_i)=a_{i+k}$, where subscripts are read "mod $n$" but this doesn't quite get us there. We either need to have some way of finding the number of disjoint cycles and proceeding from there, or we need to be able to compute the length of each disjoint cycle and then show that these lengths will all be the same.
Lastly, I do know that $\frac{nk}{gcd(n, k)} = lcm(n, k)$ and this might be the easier quantity to work with, although I haven't quite figured out how to introduce it to the problem.
Any help at all would be appreciated.