Find a proper positive factor of 48 and a proper positive multiple of 48 which add up to a perfect square. Can you prove that there are infinitely many such pairs?

Proper Factor is a factor of a number other 1 and itself.

Proper factors of 48 = 2, 3, 4, 6, 8, 12, 16, 24

Proper Multiple is a multiple other than itself.

Proper Multiples of 48 = 96, 144, 192, 240, 288, 336, 384 ….

A proper positive factor of 48 and a proper positive multiple of 48 which add up to a perfect square are:

4 + 96 = 100 = 10²

4 + 192 = 196 = 14²

16 + 240 = 256 = 16²

16 + 384 = 400 = 20²

Hence, there are infinitely many such pairs