Prove that at least one side length of a rectangle is an integer

Question

A rectangle is divided into smaller rectangles of random sizes each having at least one side of integer length. Prove that larger rectangle has at least one side of integral length.

I don't know how to approach this problem and where to start from.

Answer 1

Let $f(x,y)=\sin(\pi x)\,\sin (\pi y)$ and note that $\int_{y_0}^{y_1}\int_{x_0}^{x_1} f(x,y)\,\mathrm dx\,\mathrm dy$ is zero iff at least one of $x_1-x_0$, $y_1-y_0$ is an integer.