KöMaL 2015 A647

Let $k$ be a nonnegative integer. Prove that there are only finitely many positive integers $n$ for which there exist two disjoint sets $A$ and $B$ satisfying $A \cup B = \{1; 2; \ldots; n\}$ and $\left|\prod \limits_{a \in A} {a} - \prod\limits _{b \in B} {b}\right|=k$ .