Please comment your solutions, questions and remarks..
This riddle is not easy. I solved it only after I got a hint. I got the hint without wanting it though, and I was already thinking in the right direction; so with hindsight I say that I probably was able to solve it without the hint. Given time. And passion.
Suppose a rectangle R is divided into smaller rectangles in a way similar to that pictured below, i.e. the small rectangles cover R and they do not mutually intersect except at their boundaries. Suppose that each of the smaller rectangles has the following property: its height or length (or both) is an integer. Show that then R has the same property: the height or length (or both) of R is an integer.
I can offer hints in the comments by request. Good luck!