In general, $\displaystyle \mbox{GCD}(a,b) = \mbox{GCD}(\alpha a+\beta b,\gamma a + \eta b)$ for any matrix $\displaystyle \begin{pmatrix}
{\alpha}&{\beta}\\
{\gamma}&{\eta}
\end{pmatrix} \in \mbox{SL}_2(\mathbb{Z})$.
In general, $\displaystyle \mbox{GCD}(a,b) = \mbox{GCD}(\alpha a+\beta b,\gamma a + \eta b)$ for any matrix $\displaystyle \begin{pmatrix}
{\alpha}&{\beta}\\
{\gamma}&{\eta}
\end{pmatrix} \in \mbox{SL}_2(\mathbb{Z})$.