Question

A number is called a perfect square if it is the square of an integer. How many pairs of perfect squares differ by 495? (Order does not matter. So, the pair "16 and 9" is the same as "9 and 16".)

Answer 1

you want x^2-y^2 = 495
(x+y)(x-y) = 495
495 = 3^2*5*11
so, the factors are

1 3 5 9 11 15 33 45 55 99 165 495

so, if you have
x+y=495
x-y=1
248^2-247^2 = 495

x+y=165
x-y=3
84^2-81^2 = 495

and so on

Answer 2

please continue

Answer 3

nope. Just take the factors in pairs, as I did, working your way in from the ends.

Actually, even that's not necessary, unless you just want to verify the results. There are 12 factors, so that means there are 6 pairs like the two I showed above.

So, there are 6 pairs of numbers whose squares differ by 495.