The least non-negative remainder when \(3^{51}\) is divided by 7 is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6
A power of 3 that is one less than a multiple of 7 makes this easy:
\(3^3 = 27 = 28 - 1 \equiv -1 \pmod{7}\)
Since 51 = 3 × 17:
\(3^{51} = \left(3^3\right)^{17} \equiv (-1)^{17} = -1 \pmod{7}\)
A remainder of −1 is the same as least non-negative remainder 7 − 1 = 6.