 Estelle has two pieces of twine, one 18 feet long and the other 12 feet long. If she wants to cut them up to produce many pieces of twine that are all of the same length, with no twine left over, what is the greatest length, in feet, that she can make them?


To solve this problem, we need to find the greatest length that Estelle can cut both pieces of twine into, such that all resulting pieces are of the same length and there is no leftover twine. This is equivalent to finding the greatest common divisor (GCD) of the lengths of the two pieces of twine.

The two lengths of twine are:

- 18 feet
- 12 feet

### Step-by-step Solution

1. **List the Factors of Each Length:**

   - Factors of 18: \( 1, 2, 3, 6, 9, 18 \)
   - Factors of 12: \( 1, 2, 3, 4, 6, 12 \)

2. **Find the Common Factors:**

   The common factors of 18 and 12 are those numbers that appear in both lists:
   - Common factors: \( 1, 2, 3, 6 \)

3. **Identify the Greatest Common Factor:**

   The greatest common factor among these is \( 6 \).

### Conclusion

The greatest length that Estelle can cut both pieces of twine into, such that all pieces are of the same length and no twine is left over, is **6 feet**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What if the lengths of the twine were 24 feet and 36 feet? What would the greatest common length be then?
2. How can the Euclidean algorithm be used to find the GCD of two numbers efficiently?
3. If Estelle had a third piece of twine that was 30 feet long, how would the solution change?
4. How can finding the greatest common divisor (GCD) be useful in real-world applications other than cutting twine?
5. What is the least common multiple (LCM) of 18 and 12, and how is it calculated?

**Tip:** The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. This method is very useful when dealing with problems involving divisibility and partitioning objects into smaller equal parts.

Discover how to find the greatest length to cut twine of different lengths without leftovers. This problem involves finding the greatest common divisor (GCD) of two twine lengths, 18 feet and 12 feet. Suitable for students in Grades 6-8.

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