What are the steps to solving a greatest common factor and least common multiple problem? Provide examples

To solve problems involving the Greatest Common Factor (GCF) and Least Common Multiple (LCM), you can follow a series of systematic steps. Below are the steps, along with examples to illustrate the process.

### 1. **List the Factors (for GCF)**
   - **Steps**:
     1. Find the factors of each number.
     2. Identify the common factors.
     3. The greatest of these common factors is the GCF.
   - **Example**:
     - Find the GCF of 24 and 36.
     - **Factors of 24**: 1, 2, 3, 4, 6, 8, 12, 24
     - **Factors of 36**: 1, 2, 3, 4, 6, 9, 12, 18, 36
     - **Common Factors**: 1, 2, 3, 4, 6, 12
     - **GCF**: 12

### 2. **List the Multiples (for LCM)**
   - **Steps**:
     1. List a few multiples of each number.
     2. Identify the common multiples.
     3. The smallest of these common multiples is the LCM.
   - **Example**:
     - Find the LCM of 4 and 5.
     - **Multiples of 4**: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
     - **Multiples of 5**: 5, 10, 15, 20, 25, 30, 35, 40, ...
     - **Common Multiples**: 20, 40, ...
     - **LCM**: 20

### 3. **Prime Factorization Method (For GCF and LCM)**
   - **Steps for GCF**:
     1. Perform prime factorization for each number.
     2. Identify the common prime factors.
     3. Multiply these common primes to get the GCF.
   - **Steps for LCM**:
     1. Perform prime factorization for each number.
     2. Multiply all the primes in their highest powers.
   - **Example**:
     - Find the GCF and LCM of 18 and 24.
     - **Prime Factorization of 18**: \( 18 = 2 \times 3^2 \)
     - **Prime Factorization of 24**: \( 24 = 2^3 \times 3 \)
     - **GCF**:
       - Common Primes: 2 and 3
       - Lowest powers: \( 2^1 \) and \( 3^1 \)
       - GCF = \( 2 \times 3 = 6 \)
     - **LCM**:
       - Highest powers: \( 2^3 \) and \( 3^2 \)
       - LCM = \( 2^3 \times 3^2 = 72 \)

### 4. **Using Division Method (For GCF)**
   - **Steps**:
     1. Divide the larger number by the smaller number.
     2. Divide the divisor by the remainder.
     3. Repeat until the remainder is 0.
     4. The last divisor is the GCF.
   - **Example**:
     - Find the GCF of 48 and 18.
     - \( 48 \div 18 = 2 \) remainder 12
     - \( 18 \div 12 = 1 \) remainder 6
     - \( 12 \div 6 = 2 \) remainder 0
     - **GCF** = 6

### 5. **Using GCF to Find LCM**
   - **Steps**:
     1. Use the formula \( \text{GCF} \times \text{LCM} = \text{Product of the numbers} \) to find the LCM.
   - **Example**:
     - For numbers 18 and 24, \( \text{GCF} = 6 \).
     - \( 6 \times \text{LCM} = 18 \times 24 \)
     - \( \text{LCM} = \frac{18 \times 24}{6} = 72 \)

### Summary:
- **GCF** helps in simplifying fractions and is useful when dividing items into smaller sections.
- **LCM** is useful in finding a common time period or the least interval for events.

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Do you want details on any specific method, or have any questions?

### Related Questions:
1. What are other applications of GCF and LCM in real life?
2. How does the Euclidean algorithm work for finding the GCF?
3. Can you explain how to find the GCF and LCM for more than two numbers?
4. How do we find the GCF and LCM for algebraic expressions?
5. What is the significance of the relationship \( \text{GCF} \times \text{LCM} = \text{Product of the numbers} \)?

### Tip:
Using prime factorization is a robust method to find both GCF and LCM, especially when dealing with larger numbers.

Learn how to solve problems involving the Greatest Common Factor (GCF) and Least Common Multiple (LCM) with detailed step-by-step examples. Understand methods like listing factors, multiples, prime factorization, and using division. Suitable for students in Grades 6-8.

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