Question

When 3³³³ is divided by 11, the remainder is

• A. 1
• B. 6
• C. 5
• D. 10

Answer 1

The Correct Option is C

\(\text{We need to find the remainder when } 3^{333} \text{ is divided by } 11.\)

\(\text{Using Fermat's Little Theorem:}\)
\(\text{If } p \text{ is a prime number and } a \text{ is an integer such that } a \text{ is not divisible by } p, \text{ then } a^{p-1} \equiv 1 \pmod{p}.\)

\(\text{Here, } p = 11 \text{ and } a = 3. \text{ Fermat's Little Theorem tells us that:}\)
\(3^{10} \equiv 1 \pmod{11}.\)

\(\text{Now, we want to find } 3^{333} \pmod{11}. \text{ First, reduce the exponent modulo } 10:\)
\(333 \div 10 = 33 \text{ remainder } 3.\)

\(\text{Thus, }\)
\(3^{333} \equiv 3^3 \pmod{11}.\)

\(\text{Next, calculate } 3^3:\)
\(3^3 = 27.\)

\(\text{Now, find the remainder when 27 is divided by 11:}\)
\(27 \div 11 = 2 \text{ remainder } 5.\)
\(\text{Therefore, }\)
\(3^{333} \equiv 5 \pmod{11}.\)

\(\boxed{5}\)