Question:
If n^2 - 96 is the square of an integer, then the number of possible values of n is 4. Explain this answer in detail.

Solution:
To find the possible values of n, we need to solve the equation n^2 - 96 = m^2, where m is an integer.

Difference of Squares:
We can rewrite the equation as (n + m)(n - m) = 96. This is a factorization using the difference of squares.

Factors of 96:
To find the possible values of n, we need to find all the factor pairs of 96.

The factors of 96 are:
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.

Case 1: n + m = 96 and n - m = 1
If we choose n + m = 96 and n - m = 1, we can solve these equations simultaneously to find the value of n.

Adding the two equations, we get:
2n = 97
n = 97/2

Since n should be an integer, this case does not give us a valid solution.

Case 2: n + m = 48 and n - m = 2
If we choose n + m = 48 and n - m = 2, we can solve these equations simultaneously to find the value of n.

Adding the two equations, we get:
2n = 50
n = 25

This case gives us a valid solution where n = 25.

Case 3: n + m = 32 and n - m = 3
If we choose n + m = 32 and n - m = 3, we can solve these equations simultaneously to find the value of n.

Adding the two equations, we get:
2n = 35
n = 35/2

Since n should be an integer, this case does not give us a valid solution.

Case 4: n + m = 24 and n - m = 4
If we choose n + m = 24 and n - m = 4, we can solve these equations simultaneously to find the value of n.

Adding the two equations, we get:
2n = 28
n = 14

This case gives us a valid solution where n = 14.

Summary:
From the analysis above, we found two valid values of n, which are 25 and 14. Therefore, the number of possible values of n is 2. The correct answer is '2', not '4'.