Any product of two evens or two odds (sticking just to positives for the purpose of this problem) can be expressed as a difference of two perfect squares. 11*17=187=196-9 is an example.

A: Prove this idea.

B: Come up with a formula that gives the two perfect squares. Call the larger one a and the smaller one b.

For any two numbers a and b,

[(a+b)/2]²-[(a-b)/2]² = [(a²+2ab+b²)/4] - [(a²-2ab+b²)/4] = 4ab/4 = ab

Thus the product of a and b equals the difference of the squares of (a+b)/2 and (a-b)/2. If a and b are both even or both odd, these will be perfect squares.

Posted by Bryan on 2003-04-16 08:51:53