8.4. Does there exist a nine-digit natural number without zero digits, the remainders of the division of which by each of its digits (first, second, ..., ninth) will be pairwise distinct? Justify your answer.
Solution: According to the condition, in such a number (if it exists), all digits except zero occur only once. This means the sum of all digits is 45. By the divisibility rules for 3 and 9, the number is divisible by both 3 and 9. This means the remainders from dividing the number by these digits are 0, and therefore, they coincide.

Answer: Does not exist.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| In the assumption of the existence of such a number, some of its properties are correctly determined, but there is no solution | 2 points |
| Correct answer without justification or incorrect answer | 0 points |