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 $5$th 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?
I need to find the total number of pages in the textbook, which is a multiple of $12$ plus $11$, since each section except the epilogue is $12$ pages long. Let me call the total number of pages $p$, then I have $p = 12s + 11$, where $s$ is the number of sections. I also know that a trivia fact appears on the second-to-last page, which means that the second-to-last page is a multiple of $5$. Let me call the second-to-last page $q$, then I have $q = 5t$, where $t$ is some positive integer. I can also write $q = p - 1$, since it is one page before the last page. Combining these two equations, I get $p - 1 = 5t$, or $p = 5t + 1$. Now I have two equations for $p$, one in terms of $s$ and one in terms of $t$. I can set them equal to each other and get $12s + 11 = 5t + 1$. This is a linear equation in two variables, so I can try to solve it by finding a common multiple of $12$ and $5$. Subtracting $60$ from both sides, I get $720s + 600 = 300t$. Dividing both sides by $300$, I get $2.4s + 2 = t$. Now I need to find integer values of $s$ and $t$ that satisfy this equation. I can try different values of $s$ and see if they give integer values of $t$. For example, if $s = 1$, then $t = 4.4$, which is not an integer. If $s = 2$, then $t = 6.8$, which is also not an integer. If $s = 5$, then $t = 14$, which is finally an integer. This means that there are $5$ sections in the textbook, each with $12$ pages, except the epilogue, which has $11$ pages. Therefore, the total number of pages in the textbook is $p = 12s + 11 = 12 \times 5 + 11 = 71$. 