CW: Cutting a rectangle into smaller rectangles

This time on Cap’s Whiteboard, a quickie gem about combinatorial geometry. Is it possible to start with a rectangle, neither of whose sides has integral length, and cut it into smaller rectangular pieces, each of which has at least one integral side?

It has been said (by the mathematician Alain Connes) that you cannot understand the integers unless you understand this.

Permalink: Rectangles with an Integral Side Length