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

Options

A. a) 6
B. b) 12
C. c) 24
D. d) 48

Answer

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

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

The term \(n^2 - n\) can be factored as:
\[ n^2 - n = n(n - 1) \]
Thus, the expression becomes:
\[ (n(n - 1))(n - 2) \]

### Step 2: Analyze the Expression
Now we have:
\[ n(n - 1)(n - 2) \]
This expression represents the product of three consecutive integers: \(n\), \(n - 1\), and \(n - 2\).

### Step 3: Find the Common Divisor
The product of any three consecutive integers is always divisible by \(6\) because:
- Among any three consecutive integers, at least one is divisible by \(2\) (even).
- At least one is divisible by \(3\).

### Step 4: Check for Larger Divisors
Next, we need to check if there is a larger number that divides this expression for all \(n > 2\).

Let's evaluate the expression for some values of \(n\):
- For \(n = 3\):
\[ 3(3 - 1)(3 - 2) = 3 \cdot 2 \cdot 1 = 6 \]
- For \(n = 4\):
\[ 4(4 - 1)(4 - 2) = 4 \cdot 3 \cdot 2 = 24 \]
- For \(n = 5\):
\[ 5(5 - 1)(5 - 2) = 5 \cdot 4 \cdot 3 = 60 \]

### Step 5: Identify the Largest Common Divisor
From the calculations:
- For \(n = 3\), the result is \(6\).
- For \(n = 4\), the result is \(24\).
- For \(n = 5\), the result is \(60\).

The largest number that divides all these results is \(6\). However, we also see that \(24\) is a divisor of \(60\) and \(6\), and \(24\) is larger than \(6\).

### Conclusion
The largest number that exactly divides every number of the form \((n^2 - n)(n - 2)\) for \(n > 2\) is:
\[ \boxed{6} \]