1. **Problem statement:** We need to find how many times the digit 4 appears in the page numbers from 1 to 486. 2. **Approach:** We will count the occurrences of digit 4 in each digit place (units, tens, hundreds) separately and then sum them. 3. **Counting 4s in the units place:** - Pages go from 1 to 486. - Every 10 pages, the units digit cycles through 0 to 9. - Number of full cycles: $\lfloor \frac{486}{10} \rfloor = 48$. - Each cycle has one '4' in the units place. - So, units place 4s = $48$. - Check the last partial cycle (pages 481 to 486): pages 484 has units digit 4, so add 1. - Total units place 4s = $48 + 1 = 49$. 4. **Counting 4s in the tens place:** - Tens digit changes every 10 pages. - For tens digit to be 4, page numbers must be from 40 to 49, 140 to 149, 240 to 249, 340 to 349, 440 to 449. - Each such block has 10 pages. - Count how many such blocks are within 1 to 486: - 40-49: 10 pages - 140-149: 10 pages - 240-249: 10 pages - 340-349: 10 pages - 440-449: 10 pages - Total tens place 4s = $5 \times 10 = 50$. 5. **Counting 4s in the hundreds place:** - Hundreds digit is 4 for pages 400 to 486. - Number of pages = $486 - 400 + 1 = 87$. - So, hundreds place 4s = 87. 6. **Total number of times digit 4 appears:** $$49 + 50 + 87 = 186$$ **Final answer:** The digit 4 is used 186 times to number the pages from 1 to 486.