685. For what base of the numeral system does the following rebus have a solution

$$
\begin{aligned}
& \text { KITO } \\
& + \text { KIOTO } \\
& \hline \text { TOKIO, }
\end{aligned}
$$

where identical letters represent identical digits, and different letters represent different digits? Find all solutions to the rebus.
$\triangle$ From the last column, it is clear that the digit O is equal to zero.

From the second-to-last column $2 \mathrm{~T}=$ I or $2 \mathrm{~T}=$ I $+d$, where $d$ is the base of the numeral system. But since in the third column the sum is equal to K, not I, only the second option remains.

We obtain the system of equations:

$$
\left\{\begin{array}{l}
2 \mathrm{~T}=\mathrm{U}+d \\
\mathrm{U}+1=\mathrm{K} \\
\mathrm{K}+\mathrm{U}=d \\
\mathrm{~K}+1=\mathrm{T}
\end{array}\right.
$$

140

Since this is a system of four linear equations with four unknowns, it can be solved. In this case, it is quite easy to do.

Answer: $4350_{7}+43050_{7}=50430_{7}$.