jmc

To find the base $7$ representation of $777_{10}$, we first write $777$ as the sum of powers of $7$. To begin, we find that the largest power of $7$ that is less than $777$ is $7^3=343$. The largest multiple of $343$ that is less than $777$ is $2\cdot 343=686$, so we have $777=2\cdot 343+91$. We then consider the remainder $91$. The largest power of $7$ that is less than $91$ is $7^2=49$ and the largest multiple of $49$ that is less than $91$ is $1\cdot 49=49$. This leaves us with $91-49=42$, which can be expressed as $6\cdot 7^1$. Thus, we have $$777=2\cdot 7^3+1\cdot 7^2+6\cdot 7^1+0\cdot 7^0.$$Our base $7$ representation of $777_{10}$ is then $2160_7$. The sum of the digits of this number is $2+1+6+0=\boxed{9}$.