The number 97³0} - 14³0} is divisible by :

Correct Answer is : Both 37 and 83

Correct Answer is : Both 37 and 83

Test Index

UPSC CDS 1 2023 Math Paper

Question : 32 of 100

Marks: +1, -0

The number

97^{30} - 14^{30}

is divisible by :

Solution:

Concept:

For an even integer

n

, the expression

a^{n} - b^{n}

is always divisible by both

(a - b)

and

(a + b)

Explanation:

We have

97^{30} - 14^{30}

Here

a = 97

b = 14

, and

n = 30

, which is even.

Therefore, the number is divisible by

(97 - 14) = 83

and by

(97 + 14) = 111

Now,

111 = 37 \times 3

, so it is also divisible by

37

Thus

97^{30} - 14^{30}

is divisible by both

37

and

83

Answer:

Both 37 and 83 (Option C)

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