8.2. Find the largest four-digit number in which all digits are different and which is divisible by any of its digits (don't forget to explain why it is the largest).
Answer: 9864.

Firstly, the desired number cannot have the form $\overline{987 a}$, because divisibility by the digit 7 would mean that $a$ is 0 or 7. This means the desired number is smaller. Secondly, consider numbers of the form $\overline{986 a}$. Divisibility by the digit 9 would mean that $9+8+6+a=a+23$ is divisible by 9. Therefore, $a=4$. Thus, we get the number 9864. The sum of its digits is 27, so it is divisible by 9, and since the number 864 is divisible by 8, the number itself is divisible by 8. Therefore, it is divisible by 6 and 4.