Let \alpha be an m-cycle permutation. Prove that \alpha^j is m-cycle if and only if gcd(m,j) = 1.

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 (\alpha^j)^s = \alpha where s satisfies: sj+tm=1. But I struggle concluding that \alpha^j is m-cycle.

If I assume \alpha^j is m-cycle and that k|m \and\ k|j. If I then raise \alpha^k 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 \alpha^k.