Find the common factor of `(37^(57)+43^(57)) and (37^(37)+43^(37))`

Options

A. (43 - 37)
B. (43 + 37)
C. `(37^(37) +43^(37))`
D. 10

Answer

To find the common factor of \( (37^{57} + 43^{57}) \) and \( (37^{37} + 43^{37}) \), we can follow these steps:

### Step 1: Identify the expressions
We have two expressions:
1. \( A = 37^{57} + 43^{57} \)
2. \( B = 37^{37} + 43^{37} \)

### Step 2: Use the property of odd powers
We know from algebra that for any integers \( a \) and \( b \), if \( n \) is a positive odd integer, then:
\[ a^n + b^n \text{ is divisible by } a + b \]
In our case, both \( 57 \) and \( 37 \) are odd.

### Step 3: Apply the property to both expressions
For \( A \):
\[ A = 37^{57} + 43^{57} \text{ is divisible by } 37 + 43 \]
For \( B \):
\[ B = 37^{37} + 43^{37} \text{ is also divisible by } 37 + 43 \]

### Step 4: Calculate \( 37 + 43 \)
Now, we compute:
\[ 37 + 43 = 80 \]

### Step 5: Conclusion
Since both \( A \) and \( B \) are divisible by \( 80 \), the common factor of \( (37^{57} + 43^{57}) \) and \( (37^{37} + 43^{37}) \) is:
\[ \boxed{80} \]

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