The order of an element

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1. The problem statement, all variables and given/known data
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)


2. Relevant equations



3. 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!
 
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Let's say x^s = 1, where s < n/d. What must be the relationship between s and n/d?
 

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