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

Question

Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an \( x \) are possible?

Answer 1

We look at the remainders when powers of 3 are divided by 11:

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

These remainders repeat every 5 powers.

To find the remainder of \(3^{3333}\) when divided by 11, we first find the remainder of 3333 when divided by 5: \(3333 = 5 \times 666 + 3\).

Therefore,

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

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

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

Correct Answer: 5

Solution and Explanation

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