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To solve this problem, we need to find the greatest possible length that Damian can cut from both rolls of ribbon, ensuring that the lengths divide exactly into both 63 meters of pink ribbon and 105 meters of green ribbon. This is a **Greatest Common Divisor (GCD)** problem.

### Step-by-step solution:
1. **Find the GCD of 63 and 105:**
   - The prime factorization of 63: \( 63 = 3^2 \times 7 \)
   - The prime factorization of 105: \( 105 = 3 \times 5 \times 7 \)
   - The common factors are \( 3 \) and \( 7 \), so the GCD is \( 3 \times 7 = 21 \).

### Final Answer:
The greatest possible length Damian can cut each piece of ribbon is **21 meters**.

Would you like a further explanation or additional details? Here are 5 related questions to explore:

1. How would the answer change if the ribbon lengths were 90 meters and 120 meters?
2. What is the method to find the GCD of two numbers using the Euclidean algorithm?
3. Could Damian have a different greatest possible length if the ribbons were different widths as well?
4. How do you apply the GCD concept to problems involving more than two numbers?
5. How does finding the Least Common Multiple (LCM) differ from finding the GCD?

**Tip:** GCD helps in problems involving equal partitions, while LCM is useful when combining things into a common framework.

Damian has two rolls of ribbon, one 63 meters and the other 105 meters in length. He wants to cut them into pieces of equal length, with no ribbon left over. What is the greatest possible length of these pieces?

Discover how to find the greatest common divisor (GCD) of two numbers to determine the maximum length of ribbon that can be cut without any leftovers. This guide provides a step-by-step approach to calculating GCD, suitable for middle school students.

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