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!!

let x be the other number.
it will became 1090 + x = M (eq.1)
and 1090x = N (eq.2)

where M and N are both perfect squares.
if we factor 1090 = 2 x 5 x 109 only.

we need the variable x to have 2 x 5 x 109 x K^2
factor to get a perfect square N.

we let x = 1090K^2

we replace x in our eq. 1.

1090 + 1090K^2 = M
1090(1 + K^2) = M

therefore we need the expression 1 + K^2 to be equal to 1090 to get the smallest perfect square of M.

1 + K^2 = 1090
K^2 = 1089
K = 33
we conclude that the smallest possible value of the other number is
1090 x 33^2 = 1187010.

Posted by nickson on 2005-06-17 10:05:56