Find the largest number, which exactly divides every number of the from `(n^(3) - n) (n - 2)` ' where n is a natural number greater than 2.

Options

A. 6
B. 12
C. 24
D. 48

Answer

To find the largest number that exactly divides every number of the form \( (n^3 - n)(n - 2) \) where \( n \) is a natural number greater than 2, we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression \( (n^3 - n)(n - 2) \).

First, we simplify \( n^3 - n \):
\[ n^3 - n = n(n^2 - 1) = n(n - 1)(n + 1) \]
Thus, we can rewrite the expression as:
\[ (n(n - 1)(n + 1))(n - 2) \]

### Step 2: Identify Factors
Now, we have:
\[ n(n - 1)(n + 1)(n - 2) \]
This expression consists of four consecutive integers: \( n, n - 1, n + 1, n - 2 \).

### Step 3: Find the Common Divisor
The product of four consecutive integers is always divisible by \( 4! = 24 \). This is because among any four consecutive integers, there will be at least one multiple of 4 and at least two multiples of 2.

### Step 4: Verify with Specific Values
To ensure that 24 is indeed the largest number that divides the expression for all \( n > 2 \), we can check a few values of \( n \):

- For \( n = 3 \):
\[ (3^3 - 3)(3 - 2) = (27 - 3)(1) = 24 \]

- For \( n = 4 \):
\[ (4^3 - 4)(4 - 2) = (64 - 4)(2) = 60 \quad \text{(divisible by 24)} \]

- For \( n = 5 \):
\[ (5^3 - 5)(5 - 2) = (125 - 5)(3) = 120 \quad \text{(divisible by 24)} \]

- For \( n = 6 \):
\[ (6^3 - 6)(6 - 2) = (216 - 6)(4) = 840 \quad \text{(divisible by 24)} \]

### Conclusion
Since 24 divides the results for \( n = 3, 4, 5, \) and \( 6 \), we can conclude that the largest number that exactly divides every number of the form \( (n^3 - n)(n - 2) \) for \( n > 2 \) is:

\[ \boxed{24} \]