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jmc

prealgebra senior

Problem

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?
Solution
Each page is assigned two numbers. We can generalize the numbers assigned to page to be the pair and for . That is, if , then we can see that page number one is assigned the numbers and . It is fairly easy to see that the units digits of and will only be the same if has a units digit of or a units digit of . Thus, we simply must count how many such there are between and . Possibilities for are: 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, and 52. Therefore, there are such pages.
Final answer
11