Are you sure this problem should be in Pre-Algebra and Algebra? This is a problem for the 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

where

So the answer is 127.