  Given a multi-EGS-group $K$ acting on the $p$-adic rooted tree, where $p$ is
any prime number, we compute the exponent of the congruence quotient $K_n= K/
\St_K(n)$ for all $n\ge 1$. The formula that we obtain for $\exp(K_n)$ only
depends on $p$, $n$ and the periodicity of $K$.
