A common factor of `(13^7 + + 11^7)` and `(13^5+11^5)` is

Options

A. 24
B. `13^5+11^5`
C. `13^2+11^2`
D. None of these

Answer

To find a common factor of \( (13^7 + 11^7) \) and \( (13^5 + 11^5) \), we can use the properties of sums of powers.

### Step-by-step solution:

1. **Identify the expressions**: We have two expressions:
- \( A = 13^7 + 11^7 \)
- \( B = 13^5 + 11^5 \)

2. **Use the property of sums of powers**: For odd powers, the common factor of \( a^n + b^n \) (where \( n \) is odd) is \( a + b \). In our case, both \( 13^7 + 11^7 \) and \( 13^5 + 11^5 \) are sums of odd powers.

3. **Calculate the common factor**:
- Since both \( 7 \) and \( 5 \) are odd, we can apply the property:
- The common factor of \( A \) and \( B \) is \( 13 + 11 \).

4. **Perform the addition**:
- \( 13 + 11 = 24 \).

5. **Conclusion**: The common factor of \( (13^7 + 11^7) \) and \( (13^5 + 11^5) \) is \( 24 \).

### Final Answer:
The common factor is \( 24 \).