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.
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Follow : http://math.stackexchange.com/questions/1105240/if-a-b-%E2%88%88-z-are-coprime-show-that-2a-3b-and-3a-5b-are-coprime – lab bhattacharjee Jan 17 '15 at 14:27
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@labbhattacharjee I'm not quite understanding how that relates to my problem. – Alexis Dailey Jan 17 '15 at 14:30
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Have you followed my answer , there ? – lab bhattacharjee Jan 17 '15 at 14:30
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It seems you are talking about the greatest common divisor. – quid Jan 17 '15 at 15:30
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$\gcd(a+b,a-b)=\gcd(a+b, a+b+a-b)=\gcd(a+b,2a)$ 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 $2$.
Olórin
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I may not understand what you intend by proof, but that is a proof in my understanding of what a proof is. – Olórin Jan 17 '15 at 14:44
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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 $2$
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