Question
Aaliyah has two pieces of wire, one 15 feet long and the other 10 feet long. If she wants to cut them up to produce many pieces of wire that are all of the same length, with no wire left over, what is the greatest length, in feet, that she can make them?
Answers
GPT-4o mini
To find the greatest length of wire pieces that Aaliyah can produce, we need to determine the greatest common divisor (GCD) of the lengths of the two pieces of wire: 15 feet and 10 feet.
The steps to find the GCD are as follows:
1. **List the factors** of each length:
- Factors of 15: \( 1, 3, 5, 15 \)
- Factors of 10: \( 1, 2, 5, 10 \)
2. **Identify the common factors** of both lengths:
- Common factors: \( 1, 5 \)
3. The **greatest common factor** is:
- \( 5 \)
Thus, the greatest length in feet that Aaliyah can make the wire pieces, ensuring that there is no wire left over, is
\[
\boxed{5}
\] feet.
The steps to find the GCD are as follows:
1. **List the factors** of each length:
- Factors of 15: \( 1, 3, 5, 15 \)
- Factors of 10: \( 1, 2, 5, 10 \)
2. **Identify the common factors** of both lengths:
- Common factors: \( 1, 5 \)
3. The **greatest common factor** is:
- \( 5 \)
Thus, the greatest length in feet that Aaliyah can make the wire pieces, ensuring that there is no wire left over, is
\[
\boxed{5}
\] feet.