KBB2 Problem 3

Suppose the Cartesian plane is tiled with infinitely many square tiles such that:

1. The square tiles must have the same size and form a chessboard-like formation.

2. Each tile corner must lie on a point with integer x-coordinate and y-coordinate

3. The tile sides need not be parallel to the X or Y axis.

Given that the points $(0,0)$ and $(3,1)$ are both tile corners (not necessarily of the same tile), determine all the possible tile sizes.