8.1. (12 points) Find the smallest natural number such that after multiplying it by 9, the result is a number written with the same digits but in some different order.
Answer: 1089.

Solution. Note that the number must start with one, otherwise multiplying by 9 would increase the number of digits. After multiplying 1 by 9, we get 9, so the original number must contain the digit 9. The number 19 does not work, so two-digit numbers do not work. Let's consider three-digit numbers. The second digit (after the one) of the original number does not exceed 1, otherwise we get an increase in the number of digits. Therefore, there are only two such numbers: $\overline{1 a 9}$, where $a=0$ or $a=1$. Both numbers 109 and 119 do not meet the requirement of the problem. Let's consider three-digit numbers of the form $\overline{1 a b 9}$, where $a=0$ or $a=1$. Among them, we need to find the smallest one that satisfies the condition of the problem. But already at $a=0$ we get the number 1089, this is the answer ( $1089 \cdot 9=9801$ ).