Find $\operatorname{ord}_{11}3$. Then use what you found to find the remainder when you divide $3^{82}$ by $11$.
Work thus far:
$\operatorname{ord}_{11}3=\ ?$ $3^1\equiv3\pmod{11}$ $3^2\equiv9\pmod{11}$ $3^3\equiv5\pmod{11}$ $3^4\equiv4\pmod{11}$ $3^5\equiv1\pmod{11}$ $\operatorname{ord}_{11}3=5$
Not sure how to use this to find the remainder through.
Thanks