Let $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be an integer matrix such that $c\equiv 0$ (mod $N$) and $ad-bc=1$.
Let $u,v\in\mathbb{Z}$ such that gcd$(u,v)=1$, $v$ is a positive divisor of $N$, and $0\leq u<v$.
In my work, I should determine whether two integers $au+bv,cu+dv$ are co-prime or not.
I think that the answer is that two integers are co-prime.
My attempt:
We need to prove that there exist integers $x,y$ such that $x(au+bv)+y(cu+dv)=1$.
Equivalently, we write this equation as $(xa+yc)u+(xb+yd)v=1$.
Since $(u,v)=1$, there exist integers $P,Q$ such that $Pu+Qv=1$.
Further we can find integers $x_1,y_1,x_2,y_2$ such that $x_1a+y_1c=P$ and $x_2b+y_2d=Q$ since $(a,c)=(b,d)=1$.
However, we do not guarantee that $x_1=x_2$ and $y_1=y_2$.
Anyone help? or is there any counter example? I think that the condition $v|N$ is not necessary to prove it. Thank you.
