If n is an odd number greater than 1, then `n(n^(2)-1)` is

Options

A. divisible by 96 always
B. divisible by 48 always
C. divisible by 24 always
D. None of these

Answer

To solve the problem, we need to analyze the expression \( n(n^2 - 1) \) where \( n \) is an odd number greater than 1.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression can be rewritten using the difference of squares:
\[ n(n^2 - 1) = n(n - 1)(n + 1) \]
Here, \( n - 1 \) and \( n + 1 \) are two consecutive even numbers since \( n \) is odd.

2. **Identifying Even Numbers**:
Since \( n \) is odd, both \( n - 1 \) and \( n + 1 \) are even. Therefore, the product \( n(n - 1)(n + 1) \) will always include at least two even numbers.

3. **Finding the Factors**:
The product of two consecutive even numbers is divisible by 4. Therefore, we can conclude:
\[ (n - 1)(n + 1) \text{ is divisible by } 4 \]

4. **Considering the Odd Factor**:
The odd number \( n \) does not contribute any additional factors of 2, but since we have two even numbers, we can conclude that the entire expression \( n(n^2 - 1) \) is divisible by:
\[ 4 \times n \]

5. **Finding Divisibility by 8**:
Among the two even numbers \( n - 1 \) and \( n + 1 \), one of them is divisible by 4 (since every second even number is divisible by 4). Thus, we can conclude that:
\[ n(n - 1)(n + 1) \text{ is divisible by } 8 \]

6. **Final Check for Divisibility by 24**:
Now we need to check if \( n(n - 1)(n + 1) \) is divisible by 3. Since \( n \) is odd, it cannot be divisible by 3, but one of \( n - 1 \) or \( n + 1 \) must be divisible by 3 (as they are consecutive integers). Therefore, we can conclude that:
\[ n(n^2 - 1) \text{ is divisible by } 3 \]

7. **Combining the Results**:
Since we have established that \( n(n^2 - 1) \) is divisible by:
- \( 8 \) (from the even numbers)
- \( 3 \) (from the consecutive integers)

We can conclude that:
\[ n(n^2 - 1) \text{ is divisible by } 24 \]

### Conclusion:
Thus, if \( n \) is an odd number greater than 1, then \( n(n^2 - 1) \) is divisible by 24.