Let (a_n)_{n \ge 1} be a sequence of positive integers satisfying the following condition: 0 < a_{n + 1} - a_n \le 2001 for each n \ge 1. Then there exists infinitely many ordered pairs (p,q) of distinct positive integers such that a_p \equiv 0 \pmod {a_q}.

[originally posted by Mladenov on TSR]