1. **Problem statement:** A principal donates books to teachers. Giving each teacher 5 books leaves 23 books behind. Giving each teacher 6 books requires 8 more books. Find the number of teachers. 2. **Define variables:** Let $t$ be the number of teachers, and $B$ be the total number of books. 3. **Form equations:** - Giving 5 books each leaves 23 books: $$B = 5t + 23$$ - Giving 6 books each requires 8 more books: $$B + 8 = 6t$$ 4. **Substitute $B$ from first equation into second:** $$5t + 23 + 8 = 6t$$ $$5t + 31 = 6t$$ 5. **Solve for $t$:** $$6t - 5t = 31$$ $$t = 31$$ 6. **Answer:** There are $31$ teachers. **Final answer:** D) 31