jmc

First, we simplify $1234(\mathrm{mod}7)$ to $1234\equiv 2(\mathrm{mod}7)$. Therefore, we have $$17n\equiv 2(\mathrm{mod}7)$$This means that $17n$ can be written in the form $7a+2$, where $a$ is an integer. So we have $17n=7a+2$.

We want to find the smallest $a$ such that $\frac{7a+2}{17}=n$ is an integer. By trying values for $a$, we find that the smallest integer $a$ which satisfies the equation is $7$. Therefore, the least value for $n$ is $\frac{51}{17}=\boxed{3}$.