I have the following problem which i can only solve half of it.

Let G = <g> be a cyclic group of order n. Prove that the order of $\displaystyle g^k$ is n/d, where d = gcd(n,k).Problem:

Since G has order n, order of g is n. i.e. $\displaystyle g^n$=1. Also, d|k, giving da = k for some integer a. Hence, $\displaystyle ({g^k})^{n/d} = g^{(nk)/d} = g^{(ndb)/d} = {(g^n)}^b = 1$.Solution:

However, i am unable to show that n/d is the order. The only theorem about order that i know is that: If r is the order of g and $\displaystyle g^m =1$, then r|k. is there any other theorems that i need to know to solve this problems. Thank You!.