## Task 12 - V00612

Edgar wrote down a side calculation during a math test so hastily that he can no longer read many of the digits.

Can you figure out the illegible digits? What is the problem?

(The symbol ? is used in place of the illegible digits).

```
? ? 5 ? ? : ? 9 = ? ? ?
1 ? ?
10?
    ? 7
    2?3
    ? ? ?
0
```
Since the division works out, the last line must read 2 ? 3, and the dividend must end in 3. The units of the divisor and quotient can only total 3 if the quotient ends in 7. Similarly, the second line becomes $1 ? 5$, i.e.,

```
? ? 5 ? 3 : ? 9 = ? ? 7
1 ? 5
-----
    1 0 ?
        ? 7
    2 ? 3
    2 ? 3
    _-_-_
```

Only for a divisor of 29 or 39 does the product with 7 result in a three-digit number between 200 and 300. However, since the product of the tens digit of the quotient with the divisor should result in a two-digit number ?7, the tens digit is 3 and the divisor is 29. Otherwise, the product would be three digits. Thus, by multiplying 29 by 7 and 3, we get

```
? ? 5 ? 3 : 2 9 = ? 3 7
1 ? 5
-----
    1 0 ?
        8 7
    20 3
    20 3
    -----
```

The line 10? becomes 107 and the dividend becomes ??573. At the same time, the quotient must start with 5, otherwise the second line $1 ? 5$ would not be possible. Substitution and reverse multiplication yield the unique solution:

```
15 57 3:2 9=537
14 5
    1 0 7
        8 7
    -----
        2 0 3
        2 0 3
```