CAT 2024 Slot 2 Vedic Tables G Strategy

Finding the Remainder of \( 3^{333} \div 11 \) Using Cyclicity

Step 1: Identify the cycle of \( 3^n \mod 11 \)

\[ \begin{align*} 3^1 &\equiv 3 \mod 11 \\ 3^2 &\equiv 9 \mod 11 \\ 3^3 &\equiv 27 \equiv 5 \mod 11 \\ 3^4 &\equiv 15 \equiv 4 \mod 11 \\ 3^5 &\equiv 12 \equiv 1 \mod 11 \end{align*} \] The powers of 3 modulo 11 repeat every 5 terms.

Step 2: Reduce the exponent using the cycle

Since the cycle length is 5, we compute: \[ 333 \mod 5 = 3 \Rightarrow 3^{333} \equiv 3^3 \mod 11 \] From our earlier list: \[ 3^3 \equiv 5 \mod 11 \]

✅ Final Answer:

\[ \boxed{5} \]