5. Let $n$ be an arbitrary number written with 2000 digits and divisible by 9. Let the sum of the digits of the number $n$ be denoted by $a$, the sum of the digits of $a$ be denoted by $b$, and the sum of the digits of $b$ be denoted by $c$. Determine the number $c$.
Solution. The sum of the digits of the number $n$ is divisible by 9 and is not greater than $2000 \cdot 9=18000$, i.e., $a \leq 18000$. The sum of the digits $b$ of the number $a$ is divisible by 9 and is not greater than 36 (for $a=9999$ it is equal to 36), i.e., $b \leq 36$. The sum of the digits $c$ of the number $b$ is divisible by 9 and is not greater than 9. Therefore, $c=9$.

## Eighth Section