Lesson Description
This lesson teaches you how to determine the greatest common factor (GCF) of numbers using listing factors, prime factorization, and the ladder method. The GCF is useful for simplifying fractions, factoring expressions, and solving word problems.
Learning Objectives
- Define greatest common factor (GCF).
- Calculate GCF using listing, prime factorization, and ladder method.
- Apply GCF in simplifying fractions and solving word problems.
Key Terms
- Factor: A number that divides another number evenly.
- Greatest Common Factor (GCF): The largest factor shared by two or more numbers.
- Prime Factorization: Expressing a number as a product of prime numbers.
Example 1: GCF by Listing Factors
- Find the GCF of 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Greatest common factor: 6 → GCF = 6.
Example 2: GCF by Prime Factorization
- Find GCF of 24 and 36.
- Prime factors: 24 = 2³ × 3, 36 = 2² × 3²
- Take lowest powers of common primes: 2² × 3 = 12 → GCF = 12.
Example 3: GCF Using Ladder Method
- Find GCF of 30 and 45.
- Divide both numbers by smallest prime factor that divides both: 3 → 30 ÷ 3 = 10, 45 ÷ 3 = 15
- Divide again by 5 → 10 ÷ 5 = 2, 15 ÷ 5 = 3
- Multiply divisors: 3 × 5 = 15 → GCF = 15.
Practice Problems
- Find GCF of 16 and 28.
- Find GCF of 20 and 30.
- Find GCF of 24, 36, and 60.
- Simplify 12/18 using GCF.
Tips & Common Mistakes
- Do not confuse GCF with LCM (least common multiple).
- Check your prime factorization carefully for accuracy.
- For multiple numbers, always find common factors of all numbers.
Summary
The greatest common factor is the largest number that divides two or more numbers evenly. Using listing, prime factorization, or the ladder method helps simplify fractions and solve real-world problems efficiently.
Challenge Problems
- Find the GCF of 48, 60, and 72 using prime factorization.
- Simplify 36/48 using the GCF.
- Apply GCF to factor 18x² + 24x completely.