For all `n in N, 7^(2n)-48n-1` is divisible by

Options

A. 25
B. 26
C. 1234
D. 2304

Answer

To solve the problem of determining for which numbers \( 7^{2n} - 48n - 1 \) is divisible, we will use mathematical induction and properties of binomial expansion.

### Step-by-Step Solution:

1. **Define the Expression**: Let \( a = 7^{2n} - 48n - 1 \).

2. **Rewrite the Expression**: We can express \( 7^{2n} \) as \( (7^2)^n = 49^n \). Therefore, we have:
\[ a = 49^n - 48n - 1 \]

3. **Rearranging the Expression**: Rewrite \( a \) as:
\[ a = 49^n - 1 - 48n \]

4. **Using Binomial Theorem**: We can apply the binomial theorem to expand \( (1 + 48)^n \):
\[ (1 + 48)^n = \sum_{k=0}^{n} \binom{n}{k} 48^k \]
This gives us:
\[ a = -1 - 48n + \sum_{k=0}^{n} \binom{n}{k} 48^k \]

5. **Simplifying the Expression**: The term \( -1 + 1 \) cancels out, leaving us with:
\[ a = \sum_{k=2}^{n} \binom{n}{k} 48^k \]

6. **Factoring Out Common Terms**: We can factor out \( 48^2 \) from the summation:
\[ a = 48^2 \left( \binom{n}{2} + \binom{n}{3} \cdot 48 + \cdots + \binom{n}{n} \cdot 48^{n-2} \right) \]

7. **Calculating \( 48^2 \)**: We know that:
\[ 48^2 = 2304 \]
Thus, we can express \( a \) as:
\[ a = 2304 \left( \text{some polynomial in } n \right) \]

8. **Conclusion on Divisibility**: Since \( a \) contains the factor \( 2304 \), we conclude that \( a \) is divisible by \( 2304 \).

### Final Answer:
The expression \( 7^{2n} - 48n - 1 \) is divisible by **2304**.