Question

\(3^{3333}\) divided by 11, then the remainder would be?

Answer 1

Correct Answer: 5

We examine the remainders of powers of 3 when divided by 11:

• \(3^1 \equiv 3 \pmod{11}\)
• \(3^2 \equiv 9 \pmod{11}\)
• \(3^3 \equiv 5 \pmod{11}\)
• \(3^4 \equiv 4 \pmod{11}\)
• \(3^5 \equiv 1 \pmod{11}\)

The remainders repeat every 5 terms.

We find the remainder when 333 is divided by 5: \(333 = 5 \times 66 + 3\).

Therefore,

\[ 3^{3333} \equiv 3^3 \equiv 27 \equiv 5 \pmod{11} \]

The remainder when \(3^{3333}\) is divided by 11 is 5.