Question

for how many positive integers n^2 + 96 is a perfect square​

Answer 1

Solution:Let n2 + 96 = x2

⇒ x2 – n2 = 96

⇒ (x – n) (x + n) = 96

⇒ both x and n must be odd or both even

on these condition the cases are

x – n = 2, x + n = 48

x – n = 4, x + n = 24

x – n = 6, x + n = 16

x – n = 8, x + n = 12

and the solution of these equations can be given as

x = 25, n = 23

x = 14, n = 10

x = 11, n = 5

x = 10, n = 2

So, the required values of n are 23, 10, 5, and 2.