A book with $53$ pages numbered $1$ to $53$ has its pages renumbered in reverse, from $53$ to $1$. For how many pages do the new page number and old page number share the same units digit?
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We have a book with 53 pages numbered from 1 to 53. The pages are renumbered in reverse order, so page 1 becomes page 53, page 2 becomes page 52, and so on. We need to find how many pages have the same units digit in both the old and new numbering systems.
Let's establish the relationship between the original and new page numbers. If p is the original page number and n is the new page number, we can see that page 1 becomes page 53, page 2 becomes page 52, and so on. Notice that the sum of the original and new page numbers is always 54. This gives us the relationship: p plus n equals 54, or n equals 54 minus p.
Now let's analyze the units digits. We need the units digit of p to equal the units digit of 54 minus p. Let u be the units digit of p. Since 54 has units digit 4, we have two cases. If u is less than or equal to 4, then the units digit of 54 minus p is 4 minus u, giving us u equals 2. If u is greater than 4, we borrow from the tens place, so the units digit becomes 14 minus u, giving us u equals 7. Therefore, the units digit must be either 2 or 7.
Now let's count all the pages from 1 to 53 that have units digit 2 or 7. For units digit 2, we have pages 2, 12, 22, 32, 42, and 52, which gives us 6 pages. For units digit 7, we have pages 7, 17, 27, 37, and 47, which gives us 5 pages. Therefore, the total number of pages where the old and new page numbers share the same units digit is 6 plus 5, which equals 11.
Let's summarize our solution. We established that the new page number equals 54 minus the original page number. For the units digits to match, we found that the original page must have units digit 2 or 7. Counting all such pages from 1 to 53, we get 6 pages with units digit 2 and 5 pages with units digit 7. Therefore, the answer is 11 pages where the old and new page numbers share the same units digit.