There are pairs of numbers whose sum and product are perfect squares. For instance, 5 + 20 = 25 and 5 x 20 = 100.

If the smallest number of such a pair is 1090, what is the smallest possible value of the other number? No computers!!

1090 = 2*5*109, so if the product of X (the number we seek) and 1090 is a perfect square, then X = 1090 * (n^2)

The sum of these two numbers, then is 1090 + 1090*(n^2) = 1090*(1+n^2) which is also to be a perfect square. But 99^2 = 1089 and so 99^2+1 = 1090, which is clearly the smallest possible multiple, and so n=99 gives the solution we want, and X=1090*1089=1187010 (which is the same number given in the previous solution)

Posted by Paul on 2005-06-15 21:09:10