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UPSC CDS I 2019 Math Paper

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Question : 1 of 100

Marks: +1, -0

What is the remainder when

(17^{29} + 19^{29})

18

?

6
2
1
0

Solution:

Concept:

Use modular arithmetic: find remainder when

17^{29} + 19^{29}

is divided by 18.

Explanation:

Step 1: Reduce each base modulo 18.

17 \equiv -1 \pmod{18}

because

18 - 1 = 17

.

19 \equiv 1 \pmod{18}

because

19 = 18 + 1

.

Step 2: Raise to power 29.

17^{29} \equiv (-1)^{29} = -1 \pmod{18}

(since 29 is odd).

19^{29} \equiv 1^{29} = 1 \pmod{18}

.

Step 3: Add the two residues.

(-1) + 1 = 0 \pmod{18}

.

Thus, the remainder is 0.

Answer:

0 (Option D).

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