if I have x and y be odd integers. Then 2gcd(x,y) = gcd(x+y,x-y). I am completely lost for what to do. I may be over think this but help would greatly be appreciated. This is what I have so far:

Let d = gcd(x,y) and e = gcd(x+y, x-y). To prove that e = 2d it suffices to show that 2d|e and e|2d. To show that 2d|e we need to show that 2d|(x-y) and 2d|(x+y). Thus e = (x+y)s + (x-y)t and d = xk + yl and 2d = 2xk + 2yl. (x-y) with x and y being always odd will always return even so x-y = 2n and (x+y) with x and y being always odd will always return even so x-y = 2m for some s,t,k,l,n and m in Z. Thus,

2n = (2xk + 2yl)z

2n = 2xkz +2ylz

n = xkz + ylz

2m = (2xk + 2yl)z

2m = 2xkz + 2ylz

m = xkz+ylz

But honestly I dont even know if this is right. I need help!! I have 2 others like this and have no clue where to go!!