For all n `inN,3*5^(2n+1)+2^(3n+1)` is divisible by (A) 19 (B) 17 (C) 23 (D) 25

Options

A. 19
B. 17
C. 23
D. 25

Explanation

To solve the problem, we need to show that \( P(n) = 3 \cdot 5^{2n+1} + 2^{3n+1} \) is divisible by a certain number for all natural numbers \( n \). We will check the values of \( P(n) \) for \( n = 1 \) and \( n = 2 \) and then find the highest common factor (HCF) of these two results to determine the number that divides both.

### Step-by-Step Solution:

1. **Define the Expression**:
We have \( P(n) = 3 \cdot 5^{2n+1} + 2^{3n+1} \).

2. **Calculate \( P(1) \)**:
\[ P(1) = 3 \cdot 5^{2 \cdot 1 + 1} + 2^{3 \cdot 1 + 1} \]
\[ = 3 \cdot 5^{3} + 2^{4} \]
\[ = 3 \cdot 125 + 16 \]
\[ = 375 + 16 = 391 \]

3. **Calculate \( P(2) \)**:
\[ P(2) = 3 \cdot 5^{2 \cdot 2 + 1} + 2^{3 \cdot 2 + 1} \]
\[ = 3 \cdot 5^{5} + 2^{7} \]
\[ = 3 \cdot 3125 + 128 \]
\[ = 9375 + 128 = 9503 \]

4. **Find the HCF of \( P(1) \) and \( P(2) \)**:
We need to find the highest common factor of \( 391 \) and \( 9503 \).

- **Prime Factorization of 391**:
- \( 391 = 17 \times 23 \)

- **Prime Factorization of 9503**:
- \( 9503 = 17 \times 559 \) (where \( 559 = 13 \times 43 \))

The common factor is \( 17 \).

5. **Conclusion**:
Since both \( P(1) \) and \( P(2) \) are divisible by \( 17 \), we conclude that \( P(n) \) is divisible by \( 17 \) for all natural numbers \( n \).

### Final Answer:
The correct option is (B) 17.