21.16. All integers are written in a row, starting from one. What digit stands at the 206,788th place?

## 21.6. Periods of Decimal Fractions and Repunits

| Let $p$ be a prime number different from 2 and 5. The length of the period of the number $p$ is the number of digits in the period of the decimal representation of the fraction $1 / p$.
21.16. Answer: the digit 7. There are exactly 9 single-digit numbers, 99-9=90 two-digit numbers, 999-99-9=900 three-digit numbers, 9000 four-digit numbers, and so on. Single-digit numbers occupy the first 9 positions in the written sequence, two-digit numbers occupy $90 \cdot 2=180$ positions, three-digit numbers occupy $900 \cdot 3=2700$ positions, four-digit numbers occupy $9000 \cdot 4=36000$ positions, and five-digit numbers occupy $90000 \cdot 5=450000$ positions. Therefore, the digit we are interested in belongs to a five-digit number.

Digits belonging to numbers with no more than four digits have numbers from 1 to $9+180+2700+36000=38889$. The difference $206788-38889=167899$ needs to be divided by 5 with a remainder: $167899=5 \cdot 33579+4$. The digit we are interested in belongs to the 33580-th five-digit number, i.e., the number 43579 (the first five-digit number is 10000). In this number, the digit we are interested in is in the $4-th$ position.