Q 9.

Correct answer: 4

Let $n^{2} - 440 = p^{2}$ , where p is any integer.

Thus, $n^{2} - p^{2} = 440$

=> (n + p) x (n - p) = 440

Thus, 440 must be product of two numbers, i.e. we need to consider factors of 440.

Now, 440 = 1 x 440 or 2 x 220 or 4 x 110 or 5 x 88 or 8 x 55 or 10 x 44 or 11 x 40 or 20 x 22.

Further, since sum of these two factors would be $(n + p) + (n - p) = 2n,$ we need to select only those in which sum of the factors is even .

Hence considering:

440 = 2 x 220 gives n = 111 , $111^{2} - 440 = 109^{2}$

440 = 4 x 110 gives n = 57 , $57^{2} - 440 = 53^{2}$

440 = 10 x 44 gives n = 27 , $27^{2} - 440 = 17^{2}$

440 = 20 x 22 gives n = 21 , $21^{2} - 440 = 1^{2}$