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A large rectangle in the plane is partitioned into smaller rectangles, each of which has either integer height or integer width (or both). Prove that the large rectangle also has this property.

I've given this several attempts, starting with adding up the diagonals of each smaller rectangle, I didn't think this was good enough for a definitive proof. I've worked from the larger in towards the smaller ones and the smaller ones out to the larger ones, every time it seems so easily intuitive but the I fail to connect words to it, any help is appreciated!