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UPSC CDS II 2025 Mathematics Paper

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Question : 11 of 100

Marks: +1, -0

What is the remainder when

(17^{25} + 19^{25})

18

?

0
1
3
9

Solution:

Concept:

Use modular arithmetic. Simplify large numbers by reducing the base modulo 18 first.

Explanation:

17 \bmod 18 = 17

and

17 \equiv -1 \pmod{18}

.

So

17^{25} \equiv (-1)^{25} = -1 \equiv 17 \pmod{18}

.

19 \bmod 18 = 1

. So

19^{25} \equiv 1^{25} = 1 \pmod{18}

.

Add the remainders:

17 + 1 = 18

.

18 \bmod 18 = 0

. Therefore the remainder is 0.

Answer:

0 (Option A)

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