Greatest Common Factor (GCF)
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without a remainder. Understanding GCF is essential in simplifying fractions, solving algebraic equations, and various mathematical applications.
Definition and Basic Concepts
For two or more integers, the GCF is the largest positive integer that divides each of the integers without leaving a remainder.
Example:
Consider the numbers 12 and 18.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Greatest common factor (GCF): 6
Methods to Find the GCF
Method 1: Listing All Factors
This is the most straightforward approach:
- List all factors of each number
- Identify the common factors
- Select the largest common factor
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(24, 36) = 12
Method 2: Prime Factorization
This method involves breaking down each number into its prime factors:
- Find the prime factorization of each number
- Identify the common prime factors
- Multiply the common prime factors (using the lowest exponent if a prime appears multiple times)
Example:
Find the GCF of 48 and 60
Prime factorization of 48: 24 × 3 = 2 × 2 × 2 × 2 × 3
Prime factorization of 60: 22 × 3 × 5 = 2 × 2 × 3 × 5
Common prime factors: 22 × 3 = 2 × 2 × 3 = 12
GCF(48, 60) = 12
Method 3: Euclidean Algorithm
This is an efficient method for finding the GCF, especially for large numbers:
- Divide the larger number by the smaller number
- Replace the larger number with the smaller number
- Replace the smaller number with the remainder
- Repeat until the remainder is 0
- The last non-zero remainder is the GCF
Example:
Find the GCF of 48 and 18
| Step | Division | Quotient | Remainder |
|---|---|---|---|
| 1 | 48 ÷ 18 | 2 | 12 |
| 2 | 18 ÷ 12 | 1 | 6 |
| 3 | 12 ÷ 6 | 2 | 0 |
The last non-zero remainder is 6.
GCF(48, 18) = 6
Method 4: Continuous Division Method
This is a variation of the Euclidean algorithm that's visually easier to follow:
Example:
Find the GCF of 105 and 45
105 = 45 × 2 + 15
45 = 15 × 3 + 0
Since the remainder is 0, the GCF is 15.
GCF(105, 45) = 15
Special Cases and Properties
Important Properties:
- GCF(a, 0) = |a| (The GCF of any number and 0 is the absolute value of that number)
- GCF(a, 1) = 1 (The GCF of any number and 1 is always 1)
- If a divides b evenly, then GCF(a, b) = a
- GCF(a, b) × LCM(a, b) = |a × b| (Product of GCF and LCM equals the product of the numbers)
GCF of More Than Two Numbers
To find the GCF of more than two numbers, you can:
- Find the GCF of the first two numbers
- Find the GCF of that result and the next number
- Continue until all numbers are processed
Example:
Find the GCF of 12, 18, and 24
Step 1: GCF(12, 18) = 6
Step 2: GCF(6, 24) = 6
Therefore, GCF(12, 18, 24) = 6
Applications of GCF
1. Simplifying Fractions
To simplify the fraction 24/36:
GCF(24, 36) = 12
24/36 = (24 ÷ 12)/(36 ÷ 12) = 2/3
2. Finding Equivalent Fractions
To find an equivalent fraction to 2/3 with denominator 15:
First, find the GCF of 3 and 15: GCF(3, 15) = 3
15 ÷ 3 = 5
Multiply both numerator and denominator by 5: (2 × 5)/(3 × 5) = 10/15
3. Simplifying Algebraic Expressions
To simplify: 8x³y² + 12x²y³
Find the GCF of the terms: 4x²y²
8x³y² + 12x²y³ = 4x²y²(2x + 3y)
Interactive GCF Calculator
Calculate the GCF of two numbers:
GCF Quiz
Question 1: What is the GCF of 18 and 24?
Question 2: What is the GCF of 45, 75, and 105?
Question 3: If GCF(a, b) = 12 and LCM(a, b) = 144, and we know that a = 36, what is b?
Question 4: What method is most efficient for finding the GCF of large numbers?
Question 5: What is the simplified form of the fraction 56/98?