Subset product problem
Let $S_0=\{1\}$. For $i = 1,...,n$, set $S_i\gets \{xy\mod b : x\in S_{i-1},y\in L_i\}$. Then just check whether $a \in S_n$.

It's obviously correct, since $S_i$ contains the residues which can be expressed as a product of elements, one each from $L_1,\ldots,L_i$.

Just using the definition of $S_i$ to compute it, you get a running time of $O(m n b)$, which is significantly better than brute force when $b$ is smaller than $n^m$.

By replacing the linear sweep by divide and conquer, you might be able to improve the running time even more, though I didn't check. (It will still have a factor of $b$ though.)