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