Question: Solve the following problems (13 – 16)
13. Mary has planted 12 pots of flowers and 15 pots of trees in her garden. She wants to arrange her plants in rows such that there will be the same number of rows of flowers and trees. What is the maximum number of rows she can create?
14. 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. **Problem 13:** Mary has 12 pots of flowers and 15 pots of trees. She wants to arrange them in the same number of rows for both flowers and trees. We need to find the maximum number of rows possible.
2. To find the maximum number of rows, we need to find the greatest common divisor (GCD) of 12 and 15 because the number of rows must divide both 12 and 15 exactly.
3. The factors of 12 are: 1, 2, 3, 4, 6, 12.
4. The factors of 15 are: 1, 3, 5, 15.
5. The common factors are: 1 and 3.
6. The greatest common factor is 3.
7. Therefore, the maximum number of rows is $$3$$.
8. To verify, the total number of pots is $$12 + 15 = 27$$.
9. Dividing the total pots by the number of rows: $$27 \div 3 = 9$$ pots per row.
---
10. **Problem 14:** John wants to distribute books in sets of equal size with no books left over. The sets can be of size 6 or 8. We need to find the minimum number of books John must have.
11. This is a problem of finding the least common multiple (LCM) of 6 and 8 because the total number of books must be divisible by both 6 and 8.
12. Prime factorization:
- 6 = $$2 \times 3$$
- 8 = $$2^3$$
13. The LCM is found by taking the highest powers of all prime factors:
- Highest power of 2 is $$2^3$$
- Highest power of 3 is $$3$$
14. So, $$\text{LCM} = 2^3 \times 3 = 8 \times 3 = 24$$.
15. Therefore, the minimum number of books John needs is $$24$$.