For how many positive numbers less than 1000 is it true that among the numbers $2,3,4,5,6,7,8$ and 9 there is exactly one that is not its divisor?

(E. Semerádová)
If a number is not divisible by 2, then it is also not divisible by 4, 6, and 8. If a number is not divisible by 3, then it is also not divisible by 6 and 9. If a number is not divisible by 4, then it is also not divisible by 8. If a number is not divisible by 6, then it is not divisible by 2 or 3. Therefore, none of the numbers 2, 3, 4, and 6 can be the only number from the given list that is not a divisor of the sought number.

A number divisible by all numbers from the given list except 5 must be a multiple of $7 \cdot 8 \cdot 9 = 504$, which is the least common multiple of the remaining numbers. The positive number less than $1000$ with this property is unique, and that is 504.

A number divisible by all numbers from the given list except 7 must be a multiple of $5 \cdot 8 \cdot 9 = 360$. The positive numbers less than 1000 with this property are two, and they are 360 and 720.

A number divisible by all numbers from the given list except 8 must be a multiple of $4 \cdot 5 \cdot 7 \cdot 9 = 1260$. There is no positive number less than 1000 with this property.

A number divisible by all numbers from the given list except 9 must be a multiple of $3 \cdot 5 \cdot 7 \cdot 8 = 840$. The positive number less than 1000 with this property is unique, and that is 840.

The numbers with the given properties are exactly four.

Evaluation. 2 points for the four correct options; 4 points for excluding other options and the quality of the commentary.