TITLE: If $m,n$ are coprime positive integers and $m-n$ is odd, then $(m-n),(m+n),m,n,$ are coprime each other?
QUESTION [1 upvotes]: If $m,n$ are coprime positive integers and $m-n$ is odd, then $(m-n),(m+n),m,n,$ are coprime each other?
How do I prove it?
Especially how do I prove $(m-n), (m+n)$ are coprime?

REPLY [0 votes]: Since $(m,n)=1$ we have $mx+ny=1$ for some $x,y\in\mathbb{Z}$ so $(m-n)x+n(x+y)=1$ so $((m-n),n)=1$, and also $(m+n)x+(y-x)n=1$ so $((m+n),n)=1$
In similar calculation you can show $(m+n),m)=((m-n),m)=1$
lastly say $((m+n),(m-n))=d$ so $d\mid 2m$ and $d\mid 2n$ so $d\mid 2(m,n)$ so but $(m,n)=1$ so $d=1,2$
If $d\ne 1$ Then $2\mid m+n$ so $2\mid m$ and $2\mid n$ but $(m,n)=1$ so $2\mid 1$ which is contradiction so $d=1$ must!
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