Determine which digit is in the 1000th place after the decimal point in the decimal expansion of the number $\frac{9}{28}$.

(M. Krejčová)

Hint. What does the decimal expansion of the given number look like?
The decimal expansion of the rational number $\frac{9}{28}$ is

$$
0.32 \overline{142857}
$$

where the repeating part, consisting of six digits, is marked with a bar.

Six goes into a thousand 166 times with a remainder of four $(1000 = 166 \cdot 6 + 4)$. There are two digits between the decimal point and the repeating part. Therefore, the digit in the 1000th place after the decimal point is the same as the second digit of the repeating part, which is the digit 4.

Note. In the solution, only the remainder of dividing 1000 by 6 is significant, and this can be determined without complete division as follows: The largest number divisible by 6 (i.e., divisible by 2 and 3) that is less than 1000 is 996. The remainder of $1000 \div 6$ is $1000 - 996 = 4$.