Let be an m-cycle permutation. Prove that is m-cycle if and only if .

I have an idea for both ways but I am getting stuck on concluding either case.

If I assume the gcd is 1 then I am able to reduce down to where s satisfies: . But I struggle concluding that is m-cycle.

If I assume is m-cycle and that . If I then raise I should be able to expand the product with respect to k and conclude that k must be 1. But I fail to see a systematic way to write out .