## Task 2 - 341212

Determine the number of all natural numbers $n$ that satisfy the following conditions (1), (2), (3):

(1) The number $n$ is divisible by 5.

(2) The number $n$ and its digit sum both contain no digit zero in their decimal representation.

(3) The digit sum of the digit sum of $n$ is 2.
I. If a natural number $n$ satisfies conditions (1), (2), and (3), then:

According to (2) and (3), the sum of the digits of $n$ is either 2 or 11. According to (1) and (2), the units digit of $n$ is 5. Therefore, the only possible sum of the digits of $n$ is 11; the sum of the remaining digits is consequently 6.

II. If a natural number has the units digit 5 and all other digits are non-zero and sum to 6, then the number satisfies conditions (1), (2), and (3).

Therefore, the number of such numbers is equal to the number of all different ordered combinations of digits that are all non-zero and sum to 6.

These are exactly the following combinations:

| All permutations of ... | Number | All permutations of ... | Number |
| :---: | :---: | :---: | :---: |
| $1,1,1,1,1,1$ | 1 | $1,1,1,2$ | 5 |
| $1,1,1,3$ | 4 | $1,1,2,2$ | 6 |
| $1,1,4$ | 3 | $1,2,3$ | 6 |
| 1,5 | 2 | $2,2,2$ | 1 |
| 2,4 | 2 | 3,3 | 1 |
| 6 | 1 |  |  |

The sum of these is the desired number, which is 32.