Patrick has two pieces of cord, one 48 feet long and the other 32 feet long. He wants to cut them up to produce many pieces of cord that are all of the same length, with no cord left over. What is the greatest length, in feet, that he can make them?


To solve this problem, we need to determine the greatest length of cord that can divide both pieces (48 feet and 32 feet) evenly, with no cord left over. This can be found by calculating the greatest common divisor (GCD) of 48 and 32.

**Step 1: Prime factorization of 48**
- \( 48 = 2^4 \times 3 \)

**Step 2: Prime factorization of 32**
- \( 32 = 2^5 \)

**Step 3: Find the GCD**
- The common prime factor between 48 and 32 is 2. The lowest power of 2 in both factorizations is \( 2^4 \).

Thus, the GCD of 48 and 32 is \( 2^4 = 16 \).

Therefore, the greatest length that Patrick can make all the pieces of cord, with no remainder, is **16 feet**.

Would you like further details or have any questions?

Here are 5 related questions:
1. What are the other common divisors of 48 and 32?
2. How would the GCD change if one cord was 64 feet long instead?
3. How do you calculate the least common multiple (LCM) of two numbers?
4. What is the difference between the GCD and the LCM of two numbers?
5. How does finding the GCD apply to real-world problems?

**Tip:** The GCD of two numbers is useful when dividing things into the largest possible equal parts without leftovers.

Patrick has two pieces of cord, one 48 feet long and the other 32 feet long. He wants to cut them up to produce many pieces of cord that are all of the same length, with no cord left over. What is the greatest length, in feet, that he can make them?

Discover how to solve a problem where two cords, one 48 feet long and the other 32 feet long, need to be cut into equal lengths without leftovers. Using number theory concepts like prime factorization and the greatest common divisor (GCD), we calculate the greatest length as 16 feet. This problem is suitable for students in Grades 5-7.

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