1. Find the smallest natural number ending with the four-digit number 2018, which is a multiple of the number 2017.
1. We will take advantage of the fact that each multiple of seven by a single-digit number ends with a different digit. We will gradually add the digits of the sought factor (to the given factor 2017) from the right, so that the result ends with the four-digit number 2018.

In the first step, we look for a digit whose multiple of seven ends with the digit 8:

![](https://cdn.mathpix.com/cropped/2024_04_17_710fcb593c33c027d301g-2.jpg?height=305&width=309&top_left_y=727&top_left_x=879)

The digit 8 is the only one among the multiples of seven by digits 0 to 9, and that is the number $28=4 \cdot 7$. Therefore, the question mark must be the digit 4. We add it, and in the first line under the line, we get $4 \cdot 2017=8068$, which indeed ends with the digit 8. In the final product, we need the digit 1 in the tens place; we get this digit along with the digit 6 in the tens place in the number 8068 only by adding the digit 5 - this time we are looking for a multiple of the number 2017 that ends with the digit 5:

$$
\begin{array}{rrrrr} 
& 20 & 1 & 7 \\
\times & * & * & ? & 4 \\
\hline & 8 & 0 & 6 & 8 \\
\cdots & * & * & 5 & \\
\cdots & * & * & & \\
\cdots & * & & & \\
\hline \cdots & 2 & 0 & 18
\end{array}
$$

This is satisfied only by the multiple $5 \cdot 7=35$. Similarly, we will add the remaining two digits - the digit 3 will be in the hundreds place and the digit 4 in the thousands place:

![](https://cdn.mathpix.com/cropped/2024_04_17_710fcb593c33c027d301g-2.jpg?height=355&width=909&top_left_y=1964&top_left_x=579)

Therefore, any number that, when multiplied by 2017, ends with the four-digit number 2018, must end with the four-digit number 4354, and thus the smallest such multiple is $2017 \cdot 4354=8782018$.

For a complete solution, award 6 points. For the correct choice of the method of determining the digits from the right, award 1 point. Each correct digit in the solution is worth 1 point, and the justification that the found number is the smallest is the final point. If the method is correct but the found number is incorrect, award no more than 4 points.