Pages Calculation

The problem asks us to find the total number of pages in a book, given that the total number of digits used to write all page numbers, starting from page 1, is 2019.

Let's break down the number of digits required based on the number of digits in the page numbers:

• 1-digit page numbers: Pages 1 through 9.
• 2-digit page numbers: Pages 10 through 99.
• 3-digit page numbers: Pages 100 through 999.
• 4-digit page numbers: Pages 1000 and beyond.

We will calculate the digits used for each range of pages until we exhaust the total number of digits (2019).

Digits for 1-Digit Pages

Pages 1 to 9 are the 1-digit numbers. The number of pages is $9 - 1 + 1 = 9$. Total digits used for these pages = $9 \text{ pages} \times 1 \text{ digit/page} = 9$ digits.

Remaining digits = $2019 - 9 = 2010$ digits.

Digits for 2-Digit Pages

Pages 10 to 99 are the 2-digit numbers. The number of pages is $99 - 10 + 1 = 90$. Total digits used for these pages = $90 \text{ pages} \times 2 \text{ digits/page} = 180$ digits.

Remaining digits = $2010 - 180 = 1830$ digits.

Digits for 3-Digit Pages

After accounting for 1-digit and 2-digit pages, we have 1830 digits remaining. These digits must be used for 3-digit or higher page numbers. 3-digit page numbers start from 100. Each 3-digit page number uses 3 digits.

Number of 3-digit pages that can be numbered with the remaining digits = $\frac{\text{Remaining digits}}{\text{Digits per page}} = \frac{1830}{3} = 610$ pages.

So, there are 610 pages that are 3-digit numbers. These pages start from page 100.

The last 3-digit page number is calculated by adding the number of 3-digit pages (610) to the first 3-digit page number (100) and subtracting 1 (because 100 is included in the count): Last 3-digit page number = $100 + 610 - 1 = 709$.

Total Number of Pages

The book has gone through all 1-digit pages (1-9), all 2-digit pages (10-99), and the first 610 of the 3-digit pages (100-709).

Total number of pages = (Number of 1-digit pages) + (Number of 2-digit pages) + (Number of 3-digit pages) Total number of pages = $9 + 90 + 610 = 709$ pages.

Let's verify the total number of digits used for 709 pages:

• Pages 1-9: 9 pages × 1 digit/page = 9 digits
• Pages 10-99: 90 pages × 2 digits/page = 180 digits
• Pages 100-709: These are $709 - 100 + 1 = 610$ pages. 610 pages × 3 digits/page = 1830 digits

Total digits = $9 + 180 + 1830 = 2019$ digits.

This matches the total number of digits given in the question.

Therefore, the total number of pages in the book is 709.