jmc

First, we simplify $1356(\mathrm{mod}22)$ to $1356\equiv 14(\mathrm{mod}22)$. Therefore, we have $$3n\equiv 14(\mathrm{mod}22)$$This means that $3n$ can be written in the form $22a+14$, where $a$ is an integer. So we have $3n=22a+14$.

We want to find the smallest $a$ such that $\frac{22a+14}{3}=n$ is an integer, which we can easily find to be $1$. Therefore, $n=\frac{22+14}{3}=\boxed{12}$.