Divisibility of \(41^{43} + 43^{43}\)
We need to find a number that divides the expression \(41^{43} + 43^{43}\).
Key Divisibility Rule
Recall the algebraic identity: If \(n\) is an odd integer, then \(a^n + b^n\) is always divisible by \(a+b\).
Applying the Rule
In the given expression, \(41^{43} + 43^{43}\):
- \(a = 41\)
- \(b = 43\)
- \(n = 43\)
Since the exponent \(n=43\) is an odd number, the expression \(41^{43} + 43^{43}\) is divisible by the sum of the bases, \(a+b\).
Calculation
Calculate the sum \(a+b\):
$ a+b = 41 + 43 = 84 $Conclusion
Therefore, \(41^{43} + 43^{43}\) is divisible by 84.
Comparing this result with the given options, we find that 84 is one of the options.
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