Greatest Common Factor Calculator
Find the largest positive integer that divides all given numbers without remainder
Calculate Greatest Common Factor
Enter Numbers (up to 15)
Example Calculation
Find GCF of 45 and 189
Method 1: List of Factors
Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 189: 1, 3, 7, 9, 21, 27, 63, 189
Common factors: 1, 3, 9
Result: GCF = 9
Method 2: Prime Factorization
45 = 3² × 5¹
189 = 3³ × 7¹
Common factors: 3² = 9
Result: GCF = 9
Method 3: Euclidean Algorithm
189 = 45 × 4 + 9
45 = 9 × 5 + 0
Result: GCF = 9
GCF Calculation Methods
List of Factors
Find all factors and compare
Simple but time-consuming for large numbers
Prime Factorization
Break into prime factors
Shows mathematical structure clearly
Euclidean Algorithm
Division and remainders
Most efficient for large numbers
Binary Algorithm
Uses binary operations
Efficient in computer algorithms
Real-World Uses
Construction: Determining optimal tile sizes for floors
Fractions: Simplifying fractions to lowest terms
Design: Creating patterns with repeating elements
Engineering: Gear ratios and mechanical systems
Understanding Greatest Common Factor (GCF)
What is the GCF?
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides all numbers in a set without leaving a remainder. It's essential for simplifying fractions, solving ratio problems, and various mathematical applications.
Key Properties
- •GCF(a, b) ≤ min(a, b) for positive integers
- •If GCF(a, b) = 1, then a and b are coprime
- •GCF × LCM = product of the two numbers
- •GCF is commutative: GCF(a, b) = GCF(b, a)
Practical Applications
Bathroom Tiling
For a bathroom floor 45×189 units, the largest square tiles without cutting would be 9×9 units.
Fraction Simplification
To simplify 45/189, divide both by GCF(45,189) = 9 to get 5/21.
Pattern Design
Creating repeating patterns with elements that align perfectly.