A friend of mine had a rather unusual pet - a perfectly rectangular piece of paper whose colour was unlike any other he had ever seen. Alas - one day he came home to find that his children had innocently attacked the paper with a pair of scissors... all he had left was a pile of assorted paper fragments.

Being a devoted animal (er.., paper) lover, he decided to reconstruct his original piece of paper from these fragments, all of which were still available. The first thing he discovered when he examined the fragments more closely was that his children were obviously budding engineers. It seems that every single one of the fragments was itself a perfect rectangle. Even more amazing, his children had obviously used a ruler, because whenever he measured the dimensions of the fragments, he found that either the height of the fragment, or its width (or sometimes both), was an exact number of centimetres long. Of course, the fragments had all been jumbled up, so he had no way of knowing which way up any given fragment would have been in the original piece of paper, but at least this gave him hope that he might be able to fit the pieces back together eventually. However, my friend got a bit sidetracked, because he suddenly realised something unexpected

Each of the small rectangles had the property that one (possible both) of its sides was a whole number of centimetres long. Using just this information, prove that the same thing must have been true of the large rectangle as well - at least one of its dimensions (possibly both) must have been a whole number of centimetres long.