I know the following result in elementary number theory:
If $a,b,c,d,m,n$ are integers such that $ad-bc=1$ and $mn\neq 0$, then $$\gcd(am+bn,cm+dn)=\gcd(m,n).$$
The hypothesis $ad-bc=1$ seems too weird to not be related to some $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$ property.
If we think of $\gcd$ as a function from $\mathbb{Z}\times\mathbb{Z}$ to $\mathbb{Z}$, then this result means that the action of $SL(2,\mathbb{Z})$ in $\mathbb{Z}\times\mathbb{Z}$ preserves the $\gcd$. Is there more to it?
that's it. It amounts to these two lemmas:
(I) $$ \gcd(-n,m) = \gcd(m,n) $$
(II) given $t \in \mathbb Z,$ we have $$\gcd(m + tn,n) = \gcd(m,n) $$