I have nothing cleverer than this already worked proof which makes use of two properties of the GCD.
Let GCD(m,n) = (m,n)
2) (m,n)=(m+kn,n) where k is an integer.
But before you go there I suggest you work on the proof yourself until you have exhausted all possibilities that you can think of.
Thanks Bobbym! You probably spent a lot of time working on these questions!
It did not take me a long time, the methods of experimental math (EM) knocked them all off in under a minute apiece. It took much longer to latex them up and try to replace concise computation with verbose classical math jargon and methods. This ain't because I am some kind of big brain, it is because EM makes short work of school problems. Funny thing is that it also works better on real world problems too, suggesting that the universe is some sort of classroom or school?!
While doing some research I came across this page,
which cites mathsisfun.com twice at the bottom.