The value of 351 mod 5 is …

Official answer: 2. Explanation: By Fermat's Little Theorem, since 5 is prime and gcd(3, 5) = 1: 35-1 34 1 mod 5 We express the exponent 51 in terms of multiples of 4: 51 = 4 × 12 + 3 Therefore: 351 = (34)12 × 33 112 × 27 mod 5 27 2 mod 5 Thus, the value of 351 mod 5 is 2.

Combinatorics

GATE CSE 2019 · Q21

NAT1 markEasy

Time00:00

The value of $3^{51} \bmod 5$ is _______.

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Solution

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Official answer: 2 (±0)

Combinatorics

By Fermat's Little Theorem, since $5$ is prime and $\gcd(3, 5) = 1$ :

3^{5-1} \equiv 3^4 \equiv 1 \pmod 5

We express the exponent $51$ in terms of multiples of $4$ :

51 = 4 \times 12 + 3

Therefore:

3^{51} = (3^4)^{12} \times 3^3 \equiv 1^{12} \times 27 \pmod 5

27 \equiv 2 \pmod 5

Thus, the value of $3^{51} \bmod 5$ is 2.

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This Combinatorics previous year question is Q21 of the GATE CSE 2019 paper. Find more GATE CSE PYQs on Combinatorics in Engineering Mathematics, or browse all GATE CSE PYQs.

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Combinatorics has 19 GATE CSE PYQs across 15 papers from 2012 to 2025. 5 of them are 1-mark NAT questions like this one, and GATE CSE 2019 has 2.

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