https://file.aiquickdraw.com/mathbot/file/1730937623076ep5yvzkd.jpg

Let's go through each part of this question step-by-step to solve it:

1. **Finding the GCF (Greatest Common Factor) of 132 and 324**:
   - The factors of 132 are: \( 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132 \)
   - The factors of 324 are: \( 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324 \)
   - The highest common factor among these is **12**.
   - So, **GCF = 12**.

2. **Finding the LCM (Least Common Multiple) of 132 and 324**:
   - Using the relationship: \( \text{LCM} = \frac{\text{Product of the two numbers}}{\text{GCF}} \)
   - Product of 132 and 324 is \( 132 \times 324 = 42768 \).
   - Thus, \( \text{LCM} = \frac{42768}{12} = 3564 \).
   - So, **LCM = 3564**.

3. **Dividing the product of the two numbers by the GCF**:
   - We already calculated the product of the two numbers as \( 132 \times 324 = 42768 \).
   - Dividing this by the GCF (12): \( \frac{42768}{12} = 3564 \).
   - So, the result is **3564**.

4. **Formulating a Conclusion**:
   - Based on parts (a)–(c), we observe that the product of the two numbers divided by the GCF is equal to the LCM.
   - Thus, the correct answer for part (d) is:
     - **"The product of the two numbers divided by the g.c.f. is the l.c.m."**

Would you like further details on any part of this solution?

---

### Related Questions

1. What is the relationship between GCF and LCM for any two numbers?
2. How would you find the GCF and LCM of numbers with prime factorization?
3. Can the GCF of two numbers ever be greater than one of the numbers?
4. Why does the product of two numbers divided by their GCF give the LCM?
5. How can you verify your LCM calculation without using the GCF?

---

### Tip

To quickly find the GCF of two numbers, use their prime factorizations and select the lowest power of each common prime factor.

Consider the following pair of numbers: {132, 324}. (a) Find the GCF. (b) Find the LCM. (c) Divide the product of the two numbers by the GCF. (d) Formulate a conclusion based on parts (a)–(c).

Solve the problem of finding the GCF and LCM for the numbers 132 and 324. Follow step-by-step instructions to calculate the GCF, find the LCM, and conclude by observing the relationship between GCF, LCM, and the product of the numbers. Ideal for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation