Chinese Remainder Theorem.
1657 and 2035 are pairwise coprime. Solving the CRT gives
So the answer is 1305722.
EDIT: I misread "greatest number" as "smallest positive integer." There are infinitely many numbers satisfying the given constraints, and there is no greatest number.
EDIT 2: Ah, it seems I made another misinterpretation. I thought you meant "dividing by" instead of "dividing." Okay so we have
This leads to
So the answer is 127.