jmc

Question

A math textbook with a double-digit number of pages is split into sections. Each section is exactly 12 pages long, with the exception of the epilogue, which is 11 pages long. Every page belongs to a section. Furthermore, on the bottom of each 5th page, a trivia fact is presented on the bottom of the page, starting from the fifth page. If a trivia fact appears on the bottom of the second-to-last page, then how many pages does the textbook have?

Accepted Answer

Suppose the book has p pages. It follows that p 11 12. Also, since the second-to-last page has a trivia fact, then p-1 is divisible by 5, so p 1 5. By the Chinese Remainder Theorem, since 11 1 5, then p 11 60. Now, p is a double-digit number, so it must be either 11 or 71. However, the epilogue itself is already 11 pages long, so it follows that there must be 71 pages in the book.