A common factor of `(41^(43)+43^(43))` and `(41^(41)+43^(41))` is ……

Options

A. `(43-41)`
B. `(41^(41)+43^(41))`
C. `(41^(43)+43^(43))`
D. `(41+43)`

Answer

To find a common factor of \( (41^{43} + 43^{43}) \) and \( (41^{41} + 43^{41}) \), we can use the property of sums of powers.

### Step-by-step Solution:

1. **Identify the expressions**: We have two expressions:
- \( A = 41^{43} + 43^{43} \)
- \( B = 41^{41} + 43^{41} \)

2. **Recognize the property of sums of powers**: The property states that if \( n \) is a positive odd integer, then \( a^n + b^n \) is divisible by \( a + b \). Here, both \( n = 43 \) and \( n = 41 \) are odd.

3. **Apply the property to both expressions**:
- For \( A \): Since \( 43 \) is odd, \( A = 41^{43} + 43^{43} \) is divisible by \( 41 + 43 \).
- For \( B \): Since \( 41 \) is also odd, \( B = 41^{41} + 43^{41} \) is also divisible by \( 41 + 43 \).

4. **Calculate \( 41 + 43 \)**:
\[ 41 + 43 = 84 \]

5. **Conclusion**: Therefore, the common factor of both \( (41^{43} + 43^{43}) \) and \( (41^{41} + 43^{41}) \) is \( 84 \).

### Final Answer:
The common factor is \( 84 \).