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UPSC NDA I 2018 Mathematics Solved Paper

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Question : 46 of 120

Marks: +1, -0

What is the greatest integer among the following by which the number 5^{5} + 7^{5} is divisible?

6
8
11
12

Solution:

Concept:

For odd exponent

n

, the sum

a^n + b^n

is always divisible by

a+b

.

Explanation:

We have

5^{5} + 7^{5}

where

n=5

(odd).

Using the identity:

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

for odd

n

.

Thus

5^{5} + 7^{5} = (5+7) \times (5^{4} - 5^{3}\cdot7 + \dots + 7^{4})

.

So the number is divisible by

5+7 = 12

.

Among the given options,

12

is the largest integer that divides

5^{5}+7^{5}

.

Checking the others:

6

also divides it, but

12

is greater;

8

and

11

do not divide it.

Answer:

12

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