Number theory — Identify a common factor shared by (41^43 + 43^43) and (41^41 + 43^41). Use the gcd property for sums with odd exponents.

Introduction / Context:
This problem probes a useful gcd identity for sums of like powers. When exponents are odd, the greatest common divisor of a^m + b^m and a^n + b^n is a^{gcd(m,n)} + b^{gcd(m,n)} (up to trivial factors). Recognizing this reduces what looks like a huge arithmetic task to a simple algebraic conclusion.

Given Data / Assumptions:

• Two numbers: N1 = 41^43 + 43^43 and N2 = 41^41 + 43^41.
• Both exponents 43 and 41 are odd.
• We seek a nontrivial common factor from the options.

Concept / Approach:
For odd exponents m, n: gcd(a^m + b^m, a^n + b^n) = a^{g} + b^{g}, where g = gcd(m, n). Here, gcd(43, 41) = 1, hence the gcd simplifies to a^1 + b^1 = a + b = 41 + 43.

Step-by-Step Solution:
1) Note that 43 and 41 are both odd primes, so gcd(43, 41) = 1.2) Apply the identity to a = 41, b = 43: gcd(41^43 + 43^43, 41^41 + 43^41) = 41^{1} + 43^{1}.3) Therefore a definitive common factor is 41 + 43 = 84.4) Among the options, this corresponds to “41 + 43.”

Verification / Alternative check:
Since both quantities are sums of odd powers, each is divisible by (41 + 43) individually (because a + b divides a^k + b^k for odd k). Hence (41 + 43) divides both, confirming it as a common factor.

Why Other Options Are Wrong:
“43 − 41” is 2, which need not divide sums of odd powers generally; “41^41 + 43^41” and “41^43 + 43^43” are the original numbers, not guaranteed as common factors; “41*43” is unrelated to the odd-power sum divisibility rule.

Common Pitfalls:
Misapplying the identity for even exponents; confusing a + b with a − b; assuming the gcd must be huge rather than a simple expression.

Final Answer:
41 + 43