Example 2 Given that $A$ is the sum of the digits of the decimal number $4568^{7777}$, $B$ is the sum of the digits of $A$, and $C$ is the sum of the digits of $B$, find $C$.


Solution: From the previous problem, we know that $A, B, C$ are congruent modulo 9. To determine the value of $C$, it is clearly necessary to study the range of values for $A, B, C$.

Since $4568^{777}<10000^{7777}$, and $10000^{7777}$ has $4 \times 7777+1=$ 31109 digits, it follows that $A<9 \times 31109=279981$. This implies $B \leqslant 2+9 \times 5$ $=47$, and thus $C \leqslant 4+9=13$.

Also, $4568^{777} \equiv 5^{777} \equiv 5^{3 \times 2522+1} \equiv(-1)^{2592} \cdot 5 \equiv 5(\bmod 9)$. Therefore, $C$.