Oscar needs to ship 14 rock CDs, 12 classical CDs, and 8 pop CDs. He can pack only one type of CD in each box, and he must pack the same number of CDs in each box. What is the greatest number of CDs Oscar can pack in each box?

To determine the greatest number of CDs Oscar can pack in each box, we need to find the greatest common divisor (GCD) of the three given numbers: 14 (rock CDs), 12 (classical CDs), and 8 (pop CDs). The GCD will tell us the largest number of CDs that can go into each box while ensuring all boxes contain the same number of CDs.

Let's find the GCD of 14, 12, and 8 by breaking down each number into its prime factors:

- 14: \( 2 \times 7 \)
- 12: \( 2^2 \times 3 \)
- 8: \( 2^3 \)

The only common prime factor among all three numbers is 2. Now, we take the lowest power of 2 present in all factorizations, which is \( 2^1 = 2 \).

Thus, the greatest number of CDs Oscar can pack in each box is **2**.

### Explanation:
- If Oscar packs 2 CDs per box, he can create the following number of boxes:
  - Rock CDs: \( 14 \div 2 = 7 \) boxes
  - Classical CDs: \( 12 \div 2 = 6 \) boxes
  - Pop CDs: \( 8 \div 2 = 4 \) boxes

Therefore, the greatest number of CDs Oscar can pack in each box is 2.

Would you like more details on the solution? Here are 5 related questions to deepen your understanding:

1. How would the solution change if Oscar could mix different types of CDs in one box?
2. What if Oscar needed to pack the CDs in boxes containing 3 CDs each?
3. Can you find the least common multiple (LCM) of 14, 12, and 8? How is it useful?
4. How many boxes would Oscar need if he packed only 1 CD per box?
5. If Oscar had more CDs of a different type, how would you calculate the new GCD?

**Tip**: To find the GCD of several numbers quickly, focus on the prime factors shared by all the numbers, using the lowest powers of those shared primes.

Learn how to find the greatest number of CDs Oscar can pack in each box using the greatest common divisor (GCD). This problem involves prime factorization of 14, 12, and 8. Suitable for students in Grades 5-7.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation