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Question

A rectangle is called cool if the number of square units in its area is equal to twice the number of units in its perimeter. A cool rectangle also must have integer side lengths. What is the sum of all the different possible areas of cool rectangles?

Accepted Answer

Let the side lengths of the rectangle be a and b. It follows that ab=4(a+b). Expanding and moving all the terms to the left hand side, ab-4a-4b=0. Adding 16 to both sides allows us to factor: a(b-4)-4(b-4)=(a-4)(b-4)=16. From this point, the pairs (a,b) that provide different areas are (5,20), (6,12), and (8,8), and the sum of the possible areas is 236.