If n is a natural numbers then `25^(2n)-9^(2n)` is always divisible by

Options

A. 16
B. 34
C. both 16 or 34
D. None of these

Answer

To solve the problem, we need to determine what `25^(2n) - 9^(2n)` is divisible by when n is a natural number. We can use the difference of squares formula to simplify the expression.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression \( 25^{2n} - 9^{2n} \).

2. **Recognize the Difference of Squares**:
The expression can be recognized as a difference of squares, which is given by the formula \( a^2 - b^2 = (a - b)(a + b) \).
Here, we can let \( a = 25^n \) and \( b = 9^n \).

3. **Apply the Difference of Squares Formula**:
Using the formula, we rewrite the expression:
\[ 25^{2n} - 9^{2n} = (25^n - 9^n)(25^n + 9^n) \]

4. **Evaluate Each Factor**:
- The first factor is \( 25^n - 9^n \).
- The second factor is \( 25^n + 9^n \).

5. **Check for Divisibility**:
- For \( n = 1 \):
\[ 25^1 - 9^1 = 25 - 9 = 16 \]
\[ 25^1 + 9^1 = 25 + 9 = 34 \]
- Therefore, we have:
\[ 25^{2n} - 9^{2n} = (16)(34) \]

6. **Determine Divisibility**:
- From the factors \( 16 \) and \( 34 \), we can conclude that \( 25^{2n} - 9^{2n} \) is divisible by both \( 16 \) and \( 34 \).

### Conclusion:
Thus, the expression \( 25^{2n} - 9^{2n} \) is always divisible by \( 16 \) and \( 34 \).