Question

$3^{3333}$ divided by 11, then the remainder would be?

Answer 1

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.