Group: Proving 'Weapon' = Euclidean Division Algorithm

Euclidean Division Algorithm:

m= qn + r  ; 0 ≤ r < n; (m, n, q, r) in Z

Apply:
Cyclic group of order n:
$Latex a^n = e$
Take any m, prove it is still within the cyclic group:
$Latex a^m = a^{qn+r} =a^{qn}.a^r =(a^n)^{q}.a^r = e^q.a^r = e.a^r =a^r $[QED]