What is the sum of the digits of the base $7$ representation of $777_{10}$?
I can use repeated division by $7$ and write down the remainders. $777 \div 7 = 111$ with a remainder of $0$. $111 \div 7 = 15$ with a remainder of $6$. $15 \div 7 = 2$ with a remainder of $1$. $2 \div 7 = 0$ with a remainder of $2$. So, $777_{10} = 2160_7$. Now, I just need to add up the digits in base $7$. $2 + 1 + 6 + 0 = 9$. 