## Task 2 - 210622

Fritz finds in an old textbook a five-digit natural number printed in an exercise. In one of these numbers, the digits in the units and tens place are no longer legible.

If the two illegible digits are each replaced by an asterisk $\left(^{*}\right)$, then the number has the form

$$
418 * *
$$

Moreover, Fritz thinks he can infer from the problem text that the five-digit number can be divided without remainder by 6, 7, and 9.

Investigate whether there is a five-digit number that could have been the number in question in the textbook and has all the mentioned divisibility properties!

State this number! Are there any other such numbers?
Assume there is such a number. Then it follows: The number is divisible by 6, so it is even; its units digit is therefore $0, 2, 4, 6$, or 8.

The number is also divisible by 9; the same must hold for its digit sum. This sum is 13 greater than the sum of its tens and units digits, i.e., 13 greater than the sum of its tens and units digits.

If we sequentially set the units digit to 0, 2, 4, 6, 8, then the tens digit is derived as shown in the following table:

| Units Digit | Sum of 13 and the Units Digit | Tens Digit |
| :---: | :---: | :---: |
| 0 | 13 | 5 |
| 2 | 15 | 3 |
| 4 | 17 | 1 |
| 6 | 19 | 8 |
| 8 | 21 | 6 |

Thus, the number sought can only be one of the numbers 41850, 41832, 41814, 41886, 41868. Of these, only 41832 is divisible by 7.

Therefore, only this number could have been in the specified place in the textbook; because it is the only one that has all the required divisibility properties and is of the form $418^{**}$, as specified in the problem.