Question

Cassie has two pieces of cable, one 12 feet long and the other 9 feet long. For a science project, she wants to cut them up to produce many pieces of cable that are all the same length, with no cable left over. What is the greatest length, in feet, that she can make them?

Answer 1

To find the greatest length that Cassie can cut the cables into, we need to determine the greatest common divisor (GCD) of the two lengths of cable: 12 feet and 9 feet.

We can use the prime factorization method to find the GCD.

1. **Factorization of 12:**
\[
12 = 2^2 \times 3^1
\]

2. **Factorization of 9:**
\[
9 = 3^2
\]

3. **Identifying common factors:**
The only common prime factor between 12 and 9 is \(3\).

4. **Finding the minimum exponent for each common factor:**
- For \(3\), the minimum exponent in the factorizations is \(1\) (from \(12 = 3^1\)).

5. **Calculating the GCD:**
\[
GCD = 3^1 = 3
\]

Thus, the greatest length that Cassie can cut the two pieces of cable into, without any leftover cable, is

\[
\boxed{3} \text{ feet}.
\]