What is the number of fives used in numbering a 260-page book?

The correct option is 56.

To find the total number of times the digit 5 is used to number a 260-page book (from page 1 to 260), we count its occurrences in each place value (units place and tens place) separately.

Step 1: Count the digit 5 in the units (ones) place
In every block of 10 pages (e.g., 1-10, 11-20, ..., 251-260), the digit 5 appears exactly once in the units place (i.e., page numbers ending in 5).
Since there are 260 pages in total, the number of such 10-page intervals is:
$\frac{260}{10}=26$
Thus, the digit 5 appears 26 times in the units place (on pages 5, 15, 25, ..., 255).

Step 2: Count the digit 5 in the tens place
The digit 5 appears in the tens place in specific ranges of page numbers:
- From 50 to 59: 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 (10 times)
- From 150 to 159: 150, 151, 152, 153, 154, 155, 156, 157, 158, 159 (10 times)
- From 250 to 259: 250, 251, 252, 253, 254, 255, 256, 257, 258, 259 (10 times)
Adding these together, the digit 5 appears in the tens place:
$10+10+10=30$ times.

Note: Page numbers like 55, 155, and 255 contain the digit 5 in both the units place and the tens place, and both occurrences are correctly counted individually in Step 1 and Step 2.

Step 3: Count the digit 5 in the hundreds place
Since the book only goes up to page 260, there are no pages where the digit 5 is in the hundreds place (which would start at page 500).

Step 4: Calculate the total occurrences
Summing the counts from all places:
$\text{Total number of fives}=26+30=56$