Using properties of powers, find a non-trivial common factor of the numbers 37^57 + 43^57 and 37^37 + 43^37, and then select the correct factor expression from the options.

Introduction / Context:
This question examines your understanding of algebraic factorisation for expressions of the form a^n + b^n. When n is odd, such expressions share a common factor a + b. Here you must apply this idea to two large numbers involving powers of 37 and 43 and identify a common factor that appears in both expressions.

Given Data / Assumptions:

• The first number is 37^57 + 43^57.
• The second number is 37^37 + 43^37.
• Exponents 57 and 37 are both odd integers.
• We look for algebraic common factors rather than computing the huge numbers explicitly.

Concept / Approach:
For any real numbers a and b, and any odd positive integer n, the expression a^n + b^n is divisible by a + b. Therefore a + b is a factor of a^57 + b^57 and also of a^37 + b^37 when both exponents are odd. Here a = 37 and b = 43. Thus 37 + 43 is a common factor of both expressions. We do not need the full greatest common divisor; it is enough to identify one common factor from the list of options.

Step-by-Step Solution:
Observe that 57 is odd, so 37^57 + 43^57 is divisible by 37 + 43. Similarly 37 is odd, so 37^37 + 43^37 is also divisible by 37 + 43. Therefore 37 + 43 is a common factor of both numbers. Compute 37 + 43 = 80, but the option is written symbolically as (43 + 37), which represents the same quantity.
Verification / Alternative check:
You can recall the factorisation a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + … + b^{n-1}) for n odd. Applying this with n = 57 and n = 37 clearly shows that a + b is a factor in both cases. There is no need to determine the full factorisation, because a single common factor is sufficient for this question.

Why Other Options Are Wrong:
Option a ( 43 - 37 = 6 ) is not guaranteed to divide a^n + b^n and does not follow from the standard identity. Option c ( 37 ) is a factor of powers of 37 but not necessarily of the sum with 43^n. Option d ( 10 ) is a divisor of 80 but is not expressed as a direct algebraic factor in terms of 37 and 43. Option e ( 1 ) is a trivial factor of every integer, but the question asks you to pick the correct specific factor expression from the list.

Common Pitfalls:
A typical error is to focus on the difference 43 - 37 instead of the sum 37 + 43, or to assume that a^n + b^n factors like a^n - b^n. Remember that a + b is the key factor for sums when the exponent is odd. Another pitfall is trying to expand or approximate the huge powers, which is unnecessary and impractical.

Final Answer:
A non trivial common factor of both numbers is (43 + 37), which equals 80.