KöMaL problem A, 2015

Number theory Difficulty 6.0 National Olympiad Prove it Hungary

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 $\displaystyle\left|\prod \limits_{a \in A} {a} - \prod\limits _{b \in B} {b}\right|=k$ .
Proposed by: Balázs Maga, Budapest
(5 pont)

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