7.3. Katya wrote a four-digit number on the board that was divisible by each of its digits without a remainder (there were no zeros in the number). Then she erased the first and last digits, and the number 89 remained on the board. What could have been written on the board initially? (Find all options and show that there are no others.)
Solution: The original number was divisible by 8. Therefore, the number formed by its last three digits is also divisible by 8. From the condition, this number has the form $\overline{89 x}$. Clearly, $x=6$ fits, and other numbers divisible by 8 do not fit into this decade. The entire number is divisible by 9, which means the sum of its digits is divisible by 9. Currently, it is $8+9+6=23$. It is 4 short of the nearest number divisible by 9, and 13 short of the next one. Therefore, this number is 4896. Since it is divisible by 8 and 9 by construction, it is also divisible by 4 (since 4 is a divisor of 8), and by 6 (due to divisibility by 8 and 9).

Criteria: Only the answer, answer with verification - 1 point.

One of the digits is found unambiguously - 3 points.

It is proven that it can only be 4896, but not verified that it fits - 7 points.