  Let $\Gamma$ be a Schottky semigroup in $\mathrm{SL}_2(\mathbf{Z})$, and for
$q\in \mathbf N$, let $\Gamma(q):=\{\gamma\in \Gamma: \gamma= e \text{ (mod
$q$)}\}$ be its congruence subsemigroup of level $q$. We prove the following
uniform congruence counting theorem with respect to the family of Euclidean
norm balls $B_R$ in $M_2(\mathbf{R})$ of radius $R$: for all $q$ with no small
prime factors, $ (\Gamma (q) \cap B_R )= c_\Gamma \frac{R^{2\delta}}{
(\mathrm{SL}_2(\mathbf{Z}/q\mathbf{Z}))} +O(q^C R^{2\delta -\epsilon})$ as
$R\to \infty$ for some $c_\Gamma >0, C>0, \epsilon>0$ which are independent of
$q$. Our technique also applies to give a similar counting result for the
continued fractions semigroup of $\mathrm{SL}_2(\mathbf{Z})$, which arises in
the study of Zaremba's conjecture on continued fractions.
