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!!
(In reply to answer
by K Sengupta)
Let the minimum value of the other number be T.
By the problem, 1090*T is a perfect square. Since 1090 does not admit any perfect square divisor, it follows that: T = 1090*(U^2), for some positive integer U.
Again, (1090 + T) is a perfect square.
-> 1090(1 + U^2) = W^2, for some integer W. Thus 1090 must divide W, so that: W = 1090*M, for some M
Therfore, we must have: 1090(1 + U^2) = (1090*M)^2
-> U^2 = 1090*(M^2 - 1)
Now, we observe that the minimum positive integer M occurs at M = 1, giving:
U^2 = sqrt(1089) = 33
Substituting this in the original expression for T, we obtain:
T = 1090*(33^2) = 1187010
Consequently, the required smallest possible value of the other number is 1187010.
Edited on November 22, 2008, 2:36 am