GATE CS 2019 Q.31 (CS Q.21): Monoids and Groups

The value of $3^{51} \pmod 5$ is ________.

Your answer

Official answer and solution

Solution

Apply Fermat's Little Theorem to compute the remainder of 3^51 modulo 5.

Since $p = 5$ is prime and $\gcd(3, 5) = 1$ , Fermat's Little Theorem states that:

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

Express the exponent $51$ in terms of multiples of $4$ :

51 = 4 \times 12 + 3

Compute $3^{51} \pmod 5$ :

3^{51} = (3^4)^{12} \times 3^3 \equiv 1^{12} \times 27 \pmod 5 \equiv 27 \pmod 5 \equiv 2

Final answer

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