Using properties of powers and number theory, find a common factor of the numbers 37^57 + 43^57 and 37^37 + 43^37.

Introduction / Context:
This question is about recognizing algebraic patterns in expressions involving large powers. Instead of trying to compute gigantic numbers like 37^57 or 43^57, we use known divisibility properties for expressions of the form a^n + b^n where n is an odd integer. This is a standard idea in number theory used in many aptitude and Olympiad style questions.

Given Data / Assumptions:

• Two numbers are given: N1 = 37^57 + 43^57 and N2 = 37^37 + 43^37.
• Both exponents 57 and 37 are odd numbers.
• We want a common factor shared by N1 and N2.
• We are given options involving simple combinations of 37 and 43.

Concept / Approach:
For any integers a and b, if n is odd, then a^n + b^n is divisible by a + b. This can be seen from the factorization of a^n + b^n when n is odd, which includes the factor (a + b). In our problem, both 37^57 + 43^57 and 37^37 + 43^37 have odd exponents, so each is divisible by 37 + 43. Therefore, the sum 37 + 43 is a common factor of both numbers.

Step-by-Step Solution:
Let a = 37 and b = 43.Observe that 57 and 37 are odd integers.For any odd exponent n, the expression a^n + b^n has a factor (a + b).Thus 37^57 + 43^57 is divisible by 37 + 43.Similarly, 37^37 + 43^37 is also divisible by 37 + 43.Therefore, 37 + 43 is a common factor of both numbers.Compute 37 + 43 = 80 for verification, though the option is given symbolically.

Verification / Alternative check:
We can check the idea on a smaller example. Let a = 2 and b = 3. Then 2^3 + 3^3 = 8 + 27 = 35, which is divisible by 2 + 3 = 5. Similarly, 2^5 + 3^5 = 32 + 243 = 275, which is also divisible by 5. This confirms the pattern for odd powers. Returning to our problem, the same algebraic principle applies to 37 and 43 with exponents 57 and 37, confirming that 37 + 43 is indeed a common factor.

Why Other Options Are Wrong:

• (43 − 37) equals 6. While 6 may or may not divide these sums, it is not guaranteed by the standard identity for a^n + b^n.
• (37 + 37) equals 74, which does not have a special role in the factorization of a^n + b^n for a = 37, b = 43.
• The number 10 is arbitrary in this context and not linked to the algebraic structure.
• 6(43 − 37) equals 36, which again has no guaranteed relation to the given expressions.

Common Pitfalls:

• Attempting to compute or estimate huge powers directly, which is not practical.
• Confusing the identities for a^n − b^n and a^n + b^n and using the wrong factorization.
• Overlooking the condition that n must be odd for a^n + b^n to have a factor (a + b).

Final Answer:
(43 + 37)