ILL 1989 Problem 20

Let $R$ be a rectangle that is the union of a finite number of rectangles $R_i,$ $1 \leq i \leq n,$ satisfying the following conditions:

(i) The sides of every rectangle $R_i$ are parallel to the sides of $R.$ (ii) The interiors of any two different rectangles $R_i$ are disjoint. (iii) Each rectangle $R_i$ has at least one side of integral length.

Prove that $R$ has at least one side of integral length.

Variant: Same problem but with rectangular parallelepipeds having at least one integral side.