It is trivial to prove that gcd(x,y) divides gcd(x+y,x-y). How is it possible to prove gcd(x+y,x-y) divides gcd(x,y)? I don´t know how to use the fact that x is odd and y is even. Can anybody help me prove the statement?
Suppose $d\,|\,\gcd (x+y,x-y)$. Then $d\,|\,(x+y+x-y)=2x$. Now, the parity assumptions tell us that both $x\pm y$ are odd so $d$ must be odd. Hence $d\,|\,2x\implies d\,|\,x$.
Can you finish from here?