cycle decomposition of n-cycle's power

[TTob]

1. The problem statement, all variables and given/known data
let a=(b_1,...,b_n) n-cycle in the permutation group S_n .
prove that the cycle decomposition of a^k consist of gcd(n,k) cycles of n/gcd(n,k) size.

3. The attempt at a solution
I know that a^k(b_i)=b_{i+k (mod n)}
how can it help me ?

[morphism]

Have you tried playing with small examples and looking for patters than you can generalize?