2. The result of dividing 57 by 111 is a number of the form $0, \ldots$ with infinitely many digits after the decimal point. What is the sum of the first 2022 digits after the decimal point?
(A) 3033
(B) 4044
(C) 5055
(D) 6066
(E) 7077
## 2. The answer is $(\mathbf{D})$.

We have $\frac{57}{111}=\frac{57 \cdot 9}{111 \cdot 9}=\frac{513}{999}$. As is known, a fraction $\frac{a}{b}$ with a denominator of the form $b=\underbrace{9 \ldots 9}_{k \text { nines }}$ has a decimal expansion that is periodic, with a period of length $k$. Moreover, if the numerator is less than the denominator, then the numerator provides exactly the digits of the period. We then obtain

$$
\frac{57}{111}=\frac{513}{999}=0.513513513 \ldots
$$

and thus each group of three consecutive digits has a sum of $5+1+3=9$. Since 2022 digits correspond to $2022 / 3$ groups of 3 digits, the answer is $\frac{2022}{3} \cdot 9=6066$.

Note. Of course, the result of the division can also be obtained by direct calculation.

![](https://cdn.mathpix.com/cropped/2024_04_17_9f6e2f6c8f4b083321c1g-07.jpg?height=537&width=577&top_left_y=1745&top_left_x=791)