Is n/d the Smallest Order for x^k in Group <x>?

[R.P.F.]

Homework Statement

A group <x> has order n. k= nq+r where 0<= r < n. Prove that the order of x^k is n/d where d = gcd (n,k)

Homework Equations

The Attempt at a Solution

I know that (x^k)^(n/d) = 1, but how do I prove that n/d is the smallest one? I tried to assume that (x^k)^(n/d-q) = 1 but could not arrive at any contradiction.

Thank you!

 

[Tedjn]

Let's say x^s = 1, where s < n/d. What must be the relationship between s and n/d?