# Proof of the Greatest Denominator [duplicate]

Can someone help me prove this. It is a proof of the sum and difference of the greatest common denominator.

Given: x and y are integers with a GCD of 1.

Prove: that the GCD of x + y and x − y is either 1 or 2.

## marked as duplicate by Mark Fantini, Hans Engler, quid♦, N. F. Taussig, Lord_FarinJan 17 '15 at 16:07

• – lab bhattacharjee Jan 17 '15 at 14:27
• @labbhattacharjee I'm not quite understanding how that relates to my problem. – Alexis Dailey Jan 17 '15 at 14:30
• Have you followed my answer , there ? – lab bhattacharjee Jan 17 '15 at 14:30
• It seems you are talking about the greatest common divisor. – quid Jan 17 '15 at 15:30

$\gcd(a+b,a-b)=\gcd(a+b, a+b+a-b)=\gcd(a+b,2a)$ which divides $2a$, but any divisor of $a$ does not divide $b$, so it divides $2$, so it is equal (modulo multiplication by $\pm 1$) to $1$ or $2$.
Since $\mathbf{GCD}(x,y)=1$ these two $\Bbb Z$ are coprime.
$$\gcd(x+y,x-y)=\gcd(x+y, x+y+x-y)=\gcd(x+y,2x)$$
This divides $2x$, and since any divisor of $x$ does not divide $y$, it must be that it divides $2\implies$ it is either $1$ or $2$