Find all four-digit numbers that have exactly five four-digit and exactly nine one-digit divisors.

( $S$. Bednářová)
The sought numbers should have divisors $1,2,3,4,5,6,7,8,9$, so they must be divisible by their least common multiple:

$$
5 \cdot 7 \cdot 8 \cdot 9=2520 \text {. }
$$

We will consider the multiples of the number 2520 and determine their four-digit divisors:

- The number 2520 is a four-digit number, but it has only two four-digit divisors, see

$$
2520=2520 \cdot 1=1260 \cdot 2 ;
$$

the nearest smaller divisor is 840.

- The next multiple of the number 2520 is 5040. This number is a four-digit number and has exactly five four-digit divisors, see

$$
5040=5040 \cdot 1=2520 \cdot 2=1680 \cdot 3=1260 \cdot 4=1008 \cdot 5 ;
$$

the nearest smaller divisor is 840.

- The next multiple of the number 2520 is 7560. This number is a four-digit number, but it has more than five four-digit divisors, see

$$
7560=7560 \cdot 1=3780 \cdot 2=2520 \cdot 3=1890 \cdot 4=1512 \cdot 5=1260 \cdot 6=1080 \cdot 7
$$

- Further multiples of the number 2520 are no longer four-digit numbers.

The only solution to the problem is the number 5040.

Evaluation. 2 points for the least common multiple of the one-digit divisors; 2 points for the step-by-step analysis of the possibilities; 2 points for the conclusion and quality of the commentary.