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UPSC CDS 1 2022 Math Paper

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Question : 2 of 100

Marks: +1, -0

41^{43} + 43^{43}

is divisible by

80
84
86
88

Solution:

Concept:

For odd

n

,

a^n + b^n

is divisible by

a + b

.

Explanation:

We need to check divisibility of

41^{43} + 43^{43}

.

Here

n = 43

, which is odd.

Using the identity

a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + \dots + ab^{n-2} + b^{n-1})

for odd

n

, we get:

41^{43} + 43^{43} = (41 + 43) \times \text{(an integer)}

= 84 \times \text{(an integer)}

.

Therefore, the expression is completely divisible by

84

.

Answer:

84

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