`(392)^n-(392)^(n-1)` is not divisible by :

Options

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

Answer

To solve the problem `(392)^n - (392)^(n-1)`, we can factor the expression. Let's go through the steps:

### Step 1: Factor the expression
We can factor out the common term `(392)^(n-1)` from both parts of the expression:
\[ (392)^n - (392)^{n-1} = (392)^{n-1} \cdot ((392) - 1) \]

### Step 2: Simplify the expression
Now, we simplify the expression:
\[ (392)^{n-1} \cdot (392 - 1) = (392)^{n-1} \cdot 391 \]

### Step 3: Analyze divisibility
Now we need to determine what this expression is not divisible by. The expression is a product of two factors: `(392)^(n-1)` and `391`.

1. **Divisibility by \(392\)**: Since \( (392)^{n-1} \) is a power of \(392\), the entire expression is divisible by \(392\).
2. **Divisibility by \(391\)**: The expression is also divisible by \(391\) since it is one of the factors.

### Step 4: Identify the prime factors
Next, we need to find the prime factorization of \(392\) and \(391\):
- \(392 = 2^3 \times 7^2\)
- \(391\) is a prime number.

### Step 5: Determine what the expression is not divisible by
Since the expression is divisible by both \(392\) and \(391\), we need to find out what it is not divisible by.

To find numbers that are not factors of the expression, we can look for prime factors that are not included in the factorization of \(392\) or \(391\).

For example, \(3\) is a prime number that does not divide \(392\) or \(391\).

### Conclusion
Thus, the expression \((392)^n - (392)^{n-1}\) is not divisible by \(3\).