If n is natural number (greater than one) then `(392)^n - (392)^(n -1)` is not divisible by :

Options

A. 56
B. 23
C. 13
D. 17

Answer

To solve the problem, we need to determine which of the given numbers is not a divisor of the expression \( (392)^n - (392)^{n-1} \), where \( n \) is a natural number greater than 1.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ (392)^n - (392)^{n-1} \]
We can factor out \( (392)^{n-1} \):
\[ (392)^{n-1} \left( 392 - 1 \right) \]

2. **Simplify the Expression**:
Now, simplify \( 392 - 1 \):
\[ 392 - 1 = 391 \]
Thus, our expression becomes:
\[ (392)^{n-1} \cdot 391 \]

3. **Factor 392**:
Next, we factor \( 392 \):
\[ 392 = 7 \times 56 \]
Therefore, we can rewrite our expression as:
\[ (7 \times 56)^{n-1} \cdot 391 \]

4. **Analyze Divisibility**:
The expression \( (7 \times 56)^{n-1} \cdot 391 \) indicates that it is divisible by \( 7^{n-1} \) and \( 56^{n-1} \). We also need to check the divisibility of \( 391 \).

5. **Check the Options**:
We need to check which of the following numbers is not a divisor of \( (392)^{n-1} \cdot 391 \):
- 56
- 23
- 13
- 17

- **Option 1: 56** - Since \( 56^{n-1} \) is part of the expression, it is divisible by 56.
- **Option 2: 23** - We need to check if 391 is divisible by 23.
\[ 391 \div 23 = 17 \quad \text{(exact division)} \]
So, it is divisible by 23.
- **Option 3: 13** - Check if 391 is divisible by 13.
\[ 391 \div 13 \approx 30.0769 \quad \text{(not an integer)} \]
So, it is not divisible by 13.
- **Option 4: 17** - Check if 391 is divisible by 17.
\[ 391 \div 17 = 23 \quad \text{(exact division)} \]
So, it is divisible by 17.

6. **Conclusion**:
The only number that is not a divisor of the expression \( (392)^n - (392)^{n-1} \) is **13**.

### Final Answer:
**The number that is not divisible by \( (392)^n - (392)^{n-1} \) is 13.**