When 96 is added to a `N^(2)`, it gives another perfect square. If N is a natural no., how many distinct values of N are possible?

Options

A. a)3
B. b)4
C. c)5
D. d)None of these

Answer

To solve the problem, we need to find natural numbers \( N \) such that when \( 96 \) is added to \( N^2 \), the result is another perfect square. We can express this mathematically as:

\[ N^2 + 96 = M^2 \]

where \( M \) is also a natural number. Rearranging this equation gives us:

\[ M^2 - N^2 = 96 \]

This can be factored using the difference of squares:

\[ (M - N)(M + N) = 96 \]

Now, we need to find pairs of factors of \( 96 \). The factors of \( 96 \) are:

\[ 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 \]

Next, we will pair these factors such that \( (M - N) \) and \( (M + N) \) correspond to each pair of factors. Let’s denote:

\[ a = M - N \quad \text{and} \quad b = M + N \]

From these equations, we can express \( M \) and \( N \) as:

\[ M = \frac{a + b}{2} \quad \text{and} \quad N = \frac{b - a}{2} \]

For \( M \) and \( N \) to be natural numbers, both \( a + b \) and \( b - a \) must be even. This means \( a \) and \( b \) must either both be even or both be odd. Since all factors of \( 96 \) are even, we will only consider the even pairs.

Now, let’s find the pairs of factors of \( 96 \):

1. \( (1, 96) \)
2. \( (2, 48) \)
3. \( (3, 32) \)
4. \( (4, 24) \)
5. \( (6, 16) \)
6. \( (8, 12) \)

Now we will check which of these pairs yield natural numbers for \( N \):

1. For \( (1, 96) \):
- \( a = 1, b = 96 \)
- \( M = \frac{1 + 96}{2} = 48.5 \) (not a natural number)

2. For \( (2, 48) \):
- \( a = 2, b = 48 \)
- \( M = \frac{2 + 48}{2} = 25 \)
- \( N = \frac{48 - 2}{2} = 23 \) (valid)

3. For \( (3, 32) \):
- \( a = 3, b = 32 \)
- \( M = \frac{3 + 32}{2} = 17.5 \) (not a natural number)

4. For \( (4, 24) \):
- \( a = 4, b = 24 \)
- \( M = \frac{4 + 24}{2} = 14 \)
- \( N = \frac{24 - 4}{2} = 10 \) (valid)

5. For \( (6, 16) \):
- \( a = 6, b = 16 \)
- \( M = \frac{6 + 16}{2} = 11 \)
- \( N = \frac{16 - 6}{2} = 5 \) (valid)

6. For \( (8, 12) \):
- \( a = 8, b = 12 \)
- \( M = \frac{8 + 12}{2} = 10 \)
- \( N = \frac{12 - 8}{2} = 2 \) (valid)

Now, we have found the valid values of \( N \):

- \( N = 23 \)
- \( N = 10 \)
- \( N = 5 \)
- \( N = 2 \)

Thus, the distinct values of \( N \) are \( 2, 5, 10, 23 \).

The total number of distinct values of \( N \) is \( 4 \).

**Final Answer: 4 distinct values of N are possible.**