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?
So when the book is renumbered, the first page will have the number 53. And the second page will have the number 52. Right. So if we look at the units digits of the original page numbers, we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3 and so on. And if we look at the units digits of the new page numbers, we have 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1 and so on. So let's make a table to help us see which page numbers have the same units digit. Ok. So the first row will be the original page numbers and the second row will be the new page numbers. Great. So we can fill in the first row without any problem. Right. Or we could say $x+1=53-x\mod 10$ is equivalent to $2x=52\mod 10$. Which is $2x=2\mod 10$. So $x=1\mod 5$. Which means that $x$ must end with a $1$ or a $6$. So, out of the numbers between $1$ and $53$, the ones that have the same units digit as their page number are $1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51$. So the new page number and old page number share the same units digit for $11$ pages.

