Task 1 - 201211
In the following schema, a digit is to be entered in each empty field so that a correctly calculated division problem is formed. In particular, none of the multi-digit numbers in the completed schema may receive a leading digit of 0.

Prove that there is exactly one entry of digits in the empty fields that meets these requirements! Determine this entry!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-2089.jpg?height=662&width=679&top_left_y=443&top_left_x=997)
If an entry of digits meets the requirements, then:

The fifth digit of the quotient is 0, since in the 9th line of the schema, two new digits of the dividend appear simultaneously.

If $x$ is the divisor, then in the 8th line, the number $8x$ appears, as it is a three-digit number, while the numbers in the 2nd, 4th, 6th, and 10th lines are four-digit, thus larger than $8x$, and are $9x$; hence, the first, second, third, and sixth digits of the quotient are 9; thus, the quotient is 999809.

Since $8x$ is a three-digit number, it follows that $8x < 999 \quad \Rightarrow \quad x < \frac{999}{8} \quad (=124.875)$

From (1), (2), and the integer property of $x$, it follows that $x = 124$. Therefore, the dividend is $999809 \cdot 124 = 123976316$, and it must be the entry

\[
\begin{aligned}
& \begin{array}{cccccccccccccccccccc}
1 & 2 & 3 & 9 & 7 & 6 & 3 & 1 & 6 & : & 1 & 2 & 4 & = & 9 & 9 & 9 & 8 & 0 & 9
\end{array} \\
& \begin{array}{llll}
1 & 1 & 1 & 6 \\
\hline & 1 & 2 & 3
\end{array} \\
& \begin{array}{cccc}
1 & 1 & 1 & 6 \\
\hline & 1 & 2 & 1
\end{array} \\
& \begin{array}{llll}
1 & 1 & 1 & 6 \\
\hline & 1 & 0 & 0
\end{array} \\
& \begin{array}{ccc}
9 & 9 & 2 \\
\cline { 1 - 2 } & & \\
& 1 & 1
\end{array} \\
& \begin{array}{llll}
1 & 1 & 1 & 6 \\
\hline
\end{array}
\end{aligned}
\]

This indeed represents a correctly calculated division problem; in particular, none of its multi-digit numbers are written with a leading digit of 0.

Thus, it is proven that exactly this entry meets the given requirements.

## Adapted from [5]