Question

Leon has two pieces of string, one 28 feet long and the other 42 feet long. For an art project, he wants to cut them up to produce many pieces of string that are all of the same length, with no string left over. What is the greatest length, in feet, that he can make them?

Answer 1

To find the greatest length that Leon can cut the strings into without leaving any string left over, we need to determine the greatest common divisor (GCD) of the two lengths: 28 feet and 42 feet.

We can use the prime factorization method to find the GCD. First, we will factor each number into its prime factors.

**Step 1: Prime factorization**
- For 28:
\[
28 = 2 \times 14 = 2 \times 2 \times 7 = 2^2 \times 7^1
\]

- For 42:
\[
42 = 2 \times 21 = 2 \times 3 \times 7 = 2^1 \times 3^1 \times 7^1
\]

**Step 2: Identify common prime factors**
Next, we identify the common prime factors between the two factorizations:
- The prime factor **2** is present in both: the minimum exponent is \(1\) (from \(2^1\) in 42).
- The prime factor **3** is only in 42.
- The prime factor **7** is present in both: the minimum exponent is \(1\) (from \(7^1\) in both).

**Step 3: Calculate the GCD**
To calculate the GCD, we take the lowest power for each common prime factor:
\[
\text{GCD} = 2^{\min(2, 1)} \times 7^{\min(1, 1)} = 2^1 \times 7^1 = 2 \times 7 = 14
\]

Thus, the greatest common divisor of 28 and 42 is 14.

**Conclusion:**
The greatest length that Leon can make the pieces of string is \( \boxed{14} \) feet.