Question: John is organizing a book club meeting. He wants to distribute a set of books to the members. Each set should have an equal number of books, with no books left over. The books can be arranged in sets of $6$ or $8$. What is the minimum number of books John needs to have?
1. **State the problem:** John wants to find the minimum number of books such that the books can be divided evenly into sets of $6$ or $8$ with no remainder.
2. **Formula and concept:** This is a problem of finding the Least Common Multiple (LCM) of $6$ and $8$. The LCM of two numbers is the smallest positive integer divisible by both numbers.
3. **Find the prime factorizations:**
- $6 = 2 \times 3$
- $8 = 2^3$
4. **Determine the LCM:**
- Take the highest powers of all prime factors:
- For $2$, the highest power is $2^3$
- For $3$, the highest power is $3$
So, $$\text{LCM}(6,8) = 2^3 \times 3 = 8 \times 3 = 24$$
5. **Interpretation:** The minimum number of books John needs is $24$ so that they can be divided into sets of $6$ or $8$ without any leftover.
**Final answer:** $24$