3. Given a 2015-digit number divisible by 9. Let the sum of its digits be $\boldsymbol{a}$, the sum of the digits of $\boldsymbol{a}$ be $\boldsymbol{b}$, and the sum of the digits of $\boldsymbol{b}$ be $\boldsymbol{c}$. Find the number $\boldsymbol{c}$. (14 points).
Solution:

The sum of the digits of any number gives the same remainder when divided by 9 as the number itself.

The largest 2015-digit number consists of 2015 nines. The sum of its digits is $2015 * 9 = 18135$, i.e., it has 5 digits.

The sum of the digits of the largest 5-digit number is 45 (b).

Numbers less than 45 and divisible by 9 are 9, 18, 27, 36, and 45. The sum of their digits is 9.

## Answer: 9.

| Points | Criteria for evaluating the performance of task № 3 |
| :---: | :--- |
| $\mathbf{1 4}$ | A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are performed correctly. The correct answer is obtained. |
| $\mathbf{1 0}$ | A correct sequence of all steps of the solution is provided. There are gaps in the justification of upper estimates or possible values of the numbers $\boldsymbol{a}$ and (or) $\boldsymbol{b}$, or a computational error or typo that does not affect the further course of the solution. As a result of this error or typo, an incorrect answer may be obtained. |
| $\mathbf{5}$ | The problem is not solved, but its solution is significantly advanced, i.e.: - a significant part of the solution is performed correctly, possibly inaccurately (for example, a correct upper estimate is obtained only for the number $\boldsymbol{a}$ or only for the number $\boldsymbol{b}$); - the other part is either not performed or performed incorrectly, possibly even with logical errors (for example, finding another upper estimate). In this case, the solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not correspond to the above criteria for scoring 5 and 10 points. |