Question

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?

Answer 1

Page number 27 is the one whose order has not changed

Explanation:

Answer 2

11

Explanation:

Each page is assigned two numbers. We can generalize the numbers assigned to page x to be the pair x and 54-x for 1 ≤ x ≤ 53. That is, if x = 1, then we can see that page number one is assigned the numbers 1 and 54-1 = 53. It is fairly easy to see that the unit digits of x and 54-x will only be the same if x has a unit digit of 2 or a unit digit of 7. Thus, we simply must count how many such x there are between 1 and 54. Possibilities for x are 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, and 52. Therefore, there are 11 such pages. (Got from AoPS Alcumus Problems)