\(3^{500} \pmod 7 =\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
A power of 3 that leaves remainder 1 on division by 7 makes the large power easy:
\(3^3 = 27 = 28 - 1 \equiv -1 \pmod 7\)
\(3^6 = (3^3)^2 \equiv (-1)^2 = 1 \pmod 7\)
Dividing the exponent by 6: 500 = 6 × 83 + 2. So
\(3^{500} = (3^6)^{83}\times 3^2\)
\(\equiv 1^{83}\times 9 \pmod 7\)
\(\equiv 9 \equiv 2 \pmod 7\)