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Possible Duplicate:
Proving two gcd's equal

Let $a,b,c,d,x,y$ be integers with $\gcd(x,y)=1$ and $$ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc = 1. $$ I have come across the assertion that $(ax+by)$ and $(cx+dy)$ must be relatively prime, but I don't see why.

$gcd(x,y)=1$ means there are $C,D$ such that $$Cx+Dy=1$$

I want to find $A,B$ such that $$A(ax+by) + B(cx+dy) = 1.$$

I've tried expanding and regrouping the terms in the LHS in different ways to try to use what I've got, but I'm stuck.

Can someone please help me out?

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    *Exact* duplicate of said question. Answers are duplicates too.2012-04-14

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