jmc

To convert to base 8, we realize $8^5>6543_{10}>8^4$. So, we can tell that $6543_{10}$ in base eight will have five digits. $8^4=4096$, which can go into 6543 only one time at most, leaving $6543-1\cdot 4096=2447$ for the next four digits. $8^3=512$ goes into 2447 four times at most, leaving us with $2447-4\cdot 512=399$. Then, $8^2=64$ goes into 399 six times at most, leaving $399-6\cdot 64=15$. Next, we have $8^1=8$, which can go into 15 one time, leaving $15-1\cdot 8=7$ for the ones digit. All together, the base eight equivalent of $6543_{10}$ is $14617_8$. We are looking for the product of the digits, which is $1\cdot 4\cdot 6\cdot 1\cdot 7=\boxed{168}$.