### Instruction: What is the sum of the digits of the base $7$ representation of $777_{10}$?### Response: I need to convert $777_{10}$ to base $7$ first.
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 $7$.
$2 + 1 + 6 + 0 = 9$.