Is there a rectangle built out of some number of smaller rectangles, so that all the smaller rectangles have equal areas but different perimeters, and so that all the side lengths of all the rectangles are whole numbers?

You want to chop a rectangle up into smaller rectangles (at least 2 of them, probably more) so that all the pieces have equal areas. This much is easy: just cut it into congruent rectangles as if you were cutting up a rectangular birthday cake. But the problem asks for more: the perimeters of all the pieces should be different. Actually there is a way to do this, shown in the figure, but there the perimeters are pretty complicated numbers involving lots of square roots. Is there a way to do this so that the perimeters are all integers (whole numbers)?