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We want to prove that the following sequence: $$0, 1n, 2n, ... , (m-1)n \; (mod\; m)$$
Is equal to the sequence:$$0,d,2d, ... ,(m'-1)d$$ In some order, repeated $d$ times. Where $d=gcd(m,n)$.
Assume $m=dm'$ and $n=dn'$. They do it in two steps:
1) Prove that the sequence $0, 1n, 2n, ... ...