I was reading a proof for a problem related to groups, where originally n=3k for generators x,y and relations $x^n=1$,$y^2=1$, and $xy=yx^2$ and it stated the following: $$(3, n)=1 \rightarrow n=3k+1$$ or $$n=3k+2$$
And thus, x=1 holds as a relation.
I understand how x=1 follows from n=3k+1 or n=3k+2=3(k+1)-1, but how does n=3k+1 or n=3k+2 follow from (3, n)=1?