Common Factor Calculator
Find all common factors of two or more numbers, including the Greatest Common Factor (GCF). Analyze factor relationships and understand how numbers share divisors.
Number Input
Common Factors
Individual Factors
Prime Factorizations
Step-by-Step Process
Methods to Find Common Factors
1. Listing Method
Find all factors of each number, then identify which factors appear in all lists. The largest common factor is the GCF.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6 → GCF = 6
2. Prime Factorization Method
Break each number into prime factors, then multiply the common prime factors (taking the lowest power of each common prime).
12 = 2² × 3
18 = 2 × 3²
GCF = 2¹ × 3¹ = 6
3. Euclidean Algorithm (for two numbers)
Repeatedly divide the larger number by the smaller number and replace the larger with the remainder until the remainder is 0. The last non-zero remainder is the GCF.
18 = 12 × 1 + 6
12 = 6 × 2 + 0
GCF = 6
How to Use This Calculator
Understanding Common Factors
What are Common Factors?
Common factors are positive integers that divide evenly into two or more numbers. They represent the shared divisors among the given numbers.
Greatest Common Factor (GCF)
The GCF, also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides all the given numbers without leaving a remainder. It's the most important common factor.
Because 6 is the largest number that divides 12, 18, and 24 evenly.
Special Cases
Coprime Numbers
Numbers whose GCF is 1 are called coprime or relatively prime. They share no common factors except 1.
One Number Divides Another
When one number divides another evenly, the smaller number is the GCF of both numbers.
Prime Numbers
Two different prime numbers are always coprime (GCF = 1) since primes have no factors other than 1 and themselves.
Powers of Same Base
For numbers like 8, 16, 32 (powers of 2), the GCF is the lowest power of the common base.
Applications of Common Factors
Mathematics
- Simplifying fractions to lowest terms
- Finding equivalent fractions
- Solving Diophantine equations
- Number theory research
Real-World Problems
- Arranging objects in equal groups
- Finding common meeting times
- Tile and pattern design
- Resource allocation problems
Engineering
- Gear ratio calculations
- Signal processing and frequencies
- Structural design with modular components
- Computer algorithm optimization
Computer Science
- Cryptographic algorithms
- Hash table sizing
- Memory allocation optimization
- Parallel processing task division
Example Calculations
Example 1: Two Numbers
Example 2: Three Numbers
Example 3: Coprime Numbers
Frequently Asked Questions
What's the difference between GCF and LCM?
GCF (Greatest Common Factor) is the largest number that divides all given numbers, while LCM (Least Common Multiple) is the smallest number that all given numbers divide into. They are related: GCF × LCM = product of the numbers (for two numbers).
Can the GCF be larger than one of the numbers?
No, the GCF cannot be larger than the smallest number in the set. The maximum possible GCF is the smallest number itself (when one number divides all others).
What if I enter the same number multiple times?
If you enter the same number multiple times, the GCF will be that number itself, and all factors of that number will be common factors. The calculator will still work correctly.
How do I simplify a fraction using GCF?
To simplify a fraction, find the GCF of the numerator and denominator, then divide both by the GCF. For example: 12/18 → GCF(12,18) = 6 → 12÷6 / 18÷6 = 2/3.