Page Numbering Digit Calculation

The problem requires finding the total number of pages in a book when the total count of digits used for page numbering is 2019.

Digit Distribution Analysis

We can calculate the digits used based on the number of digits per page number:

• 1-digit pages: Pages 1 to 9.
  • Number of pages: 9
  • Digits per page: 1
  • Total digits used: $9 \times 1 = 9$
• 2-digit pages: Pages 10 to 99.
  • Number of pages: $99 - 10 + 1 = 90$
  • Digits per page: 2
  • Total digits used: $90 \times 2 = 180$

Total digits used for pages 1 through 99 is $9 + 180 = 189$.

Calculating Remaining Pages

Given total digits = 2019.

Digits remaining after numbering the first 99 pages: $2019 - 189 = 1830$.

These remaining digits are used for pages with 3 or more digits. Assuming the book doesn't have extraordinarily long page numbers, we first check 3-digit pages.

3-digit pages: Pages 100 onwards.

• Digits per page: 3
• Number of 3-digit pages = Remaining digits / Digits per page $ \text{Number of 3-digit pages} = \frac{1830}{3} = 610 $

Determining Total Pages

The total number of pages in the book is the sum of pages up to 99 and the calculated number of 3-digit pages.

• Number of pages up to 99: 99
• Number of 3-digit pages: 610
• Total pages = $99 + 610 = 709$

Alternatively, the last page number is the first 3-digit number (100) plus the number of 3-digit pages minus 1: $100 + (610 - 1) = 100 + 609 = 709$.

Verification

Let's verify the total digits for 709 pages:

• Digits for pages 1-9: $9 \times 1 = 9$
• Digits for pages 10-99: $90 \times 2 = 180$
• Digits for pages 100-709: $(709 - 100 + 1) \times 3 = 610 \times 3 = 1830$
• Total digits: $9 + 180 + 1830 = 2019$. This matches the given number.

Therefore, the book has 709 pages.