jmc

We begin by converting $682_{10}$ into base-6. We see that $6^3=216$ is the largest power of 6 that is less than 682, and that $3\cdot 216=648$ is the largest multiple of 216 that is less than 682. This leaves us with a remainder of $682-648=34$, which we can express as $5\cdot 6^1+4\cdot 6^0$. So, $682_{10}=3\cdot 6^3+0\cdot 6^2+5\cdot 6^1+4\cdot 6^0=3054_6$. The first and last digits are 3 and 4, respectively, making the product of the two equal to $\boxed{12}$.