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UPSC NDA II 2025 Mathematics Paper

Question : 9 of 120

Marks: +1, -0

5^{99}

is divided by 13?

Solution:

Concept:

Use modular arithmetic to find the repeating cycle of powers of 5 modulo 13.

Explanation:

Compute powers of 5 modulo 13 until the result becomes 1.

5^1 \equiv 5 \pmod{13}

5^2 \equiv 25 \equiv 12 \pmod{13}

5^3 \equiv 5 \times 12 \equiv 60 \equiv 8 \pmod{13}

5^4 \equiv 5 \times 8 \equiv 40 \equiv 1 \pmod{13}

This shows the cycle repeats every 4 steps.

Reduce the exponent 99 modulo 4:

99 \div 4

gives remainder 3.

Therefore,

5^{99} \equiv 5^3 \equiv 8 \pmod{13}

So the remainder when

5^{99}

is divided by 13 is 8.

Answer:

8 (Option C)

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