Geometry problem

[borisp]

Let R be a rectangle in the plane. Suppose that R can be tiled by finitely many rectangles {R_i}, and that each of the R_i has at least one side length that is an integer.

Show that R must have at least one side length that is an integer.

(This is a famous olympiad problem, so I apologize for those that have seen it before and are consequently offended.)