**Shane wants to plant 28 marigold plants and 36 rose plants in his garden. What is the greatest number of rows possible if each row has the same number of marigold plants and the same number of rose plants. Justify your answer.**

To solve this problem, we need to find the **greatest number of rows** such that:

1. Each row has the same number of marigold plants.
2. Each row has the same number of rose plants.

This means the number of plants in each row must be a divisor of both 28 (marigold plants) and 36 (rose plants). The solution lies in finding the **greatest common divisor (GCD)** of 28 and 36.

### Step-by-step solution:

1. **Prime factorization** of 28:
   \[
   28 = 2^2 \times 7
   \]

2. **Prime factorization** of 36:
   \[
   36 = 2^2 \times 3^2
   \]

3. The **common factors** between 28 and 36 are only \(2^2\) (which is 4).

4. So, the greatest common divisor (GCD) of 28 and 36 is 4.

Thus, the **greatest number of rows possible** is 4. In each row, Shane can plant:

- \( \frac{28}{4} = 7 \) marigold plants.
- \( \frac{36}{4} = 9 \) rose plants.

### Justification:
The greatest number of rows that equally divides both 28 and 36 is the GCD, which ensures that each row has the same number of marigolds and the same number of roses.

Would you like further details on this solution, or any additional questions?

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Here are 5 related questions to expand your understanding:

1. How do you find the GCD of three numbers, such as 28, 36, and another value?
2. If Shane wanted to plant 32 marigold plants and 36 rose plants, what would be the greatest number of rows?
3. What if Shane planted 28 marigold plants and 40 rose plants? How many rows would he need?
4. How does the concept of GCD apply when arranging objects in grids or tables?
5. Can the GCD method be applied if the number of plants changes every week? How?

**Tip**: The GCD is helpful not only in gardening but also in organizing any items into equal groups efficiently!

Shane wants to plant 28 marigold plants and 36 rose plants in his garden. What is the greatest number of rows possible if each row has the same number of marigold plants and the same number of rose plants? Justify your answer.

Discover how to find the greatest number of rows possible for planting 28 marigold and 36 rose plants using the GCD method. This problem involves number theory and prime factorization, making it suitable for students in Grades 6-8. Learn step-by-step how to determine the maximum rows with equal numbers of plants.

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