Common Factor Calculator - Calculate GCF & LCM with Step-by-Step Solutions
Free common factor calculator. Calculate greatest common factor, least common multiple, and common factors of numbers with step-by-step solutions. Our calculator uses number theory principles to determine all factor relationships from any given inputs.
Last updated: October 19, 2025
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Enter 2 or more numbers separated by commas
6
Greatest Common Factor
72
Least Common Multiple
4
Total common factors
Input Numbers
Common Factors
Prime Factorization
Factor Pairs
Input numbers: 12, 18, 24
Prime factors: 12 = 2 × 2 × 3, 18 = 2 × 3 × 3, 24 = 2 × 2 × 2 × 3
All factors: 12: [1, 2, 3, 4, 6, 12], 18: [1, 2, 3, 6, 9, 18], 24: [1, 2, 3, 4, 6, 8, 12, 24]
Common factors: [1, 2, 3, 6]
Greatest Common Factor (GCF): 6
Least Common Multiple (LCM): 72
Factor Listing
List all factors and find common ones
Prime Factorization
Factor each number into primes and find common factors
Euclidean Algorithm
Use Euclidean algorithm for efficient calculation
Key Relationships:
- • GCF is the largest number that divides all given numbers
- • LCM is the smallest number that is a multiple of all given numbers
- • GCF × LCM = Product of the two numbers (for two numbers)
- • Common factors are all numbers that divide all given numbers
Common Factor Calculator Types & Features
Method used
Euclidean Algorithm
Efficiently finds the largest common divisor
Formula used
LCM = (a × b) / GCF(a, b)
Uses GCF relationship for efficient calculation
Method used
Factor Listing
Lists all common divisors
Method used
Trial Division
Systematic prime factorization
Method used
Systematic Search
Finds all factor combinations
Features
Comprehensive Analysis
Complete number theory calculations
Quick Example Result
For numbers 12, 18, 24:
GCF
6
LCM
72
Common Factors
4
How Our Common Factor Calculator Works
Our common factor calculator uses the fundamental principles of number theory to calculate all factor relationships from any given inputs. The calculation applies mathematical algorithms and number theory to determine GCF, LCM, and common factors.
The Fundamental Number Theory Formulas
GCF(a,b) = GCF(b, a mod b)LCM(a,b) = (a × b) / GCF(a,b)Common Factors = ∩(Factors of each number)Prime Factors = Trial DivisionThese formulas form the foundation of number theory analysis and allow calculation of all factor relationships from any combination of known values. They apply to both small and large numbers efficiently.
Shows the relationships between GCF, LCM, and common factors
Mathematical Foundation
Common factors are fundamental concepts in number theory that describe the relationships between numbers. They are essential for understanding divisibility, fractions, and many mathematical applications in algebra, geometry, and advanced mathematics.
- GCF is the largest number that divides all given numbers
- LCM is the smallest number that is a multiple of all given numbers
- Common factors are all numbers that divide all given numbers
- Prime factorization breaks numbers into prime components
- Euclidean algorithm efficiently finds GCF
- Factor pairs show all multiplication combinations
Sources & References
- Elementary Number Theory - David M. BurtonComprehensive introduction to number theory and factor analysis
- Introduction to the Theory of Numbers - Niven, Zuckerman, MontgomeryAdvanced number theory concepts and algorithms
- Khan Academy - Number Theory and FactorsEducational resources for understanding number theory
Need help with other number theory calculations? Check out our GCF calculator and LCM calculator.
Get Custom Calculator for Your PlatformCommon Factor Calculator Examples
Given Information:
- Numbers: 12, 18, 24
- Method: Euclidean Algorithm
- Analysis: Complete factor analysis
- Output: GCF, LCM, common factors
Calculation Steps:
- Prime factors: 12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3
- Common prime factors: 2 × 3 = 6
- GCF = 6 (largest common divisor)
- LCM = 72 (smallest common multiple)
Result: GCF = 6, LCM = 72, Common Factors = [1, 2, 3, 6]
The numbers 12, 18, and 24 have a greatest common factor of 6 and least common multiple of 72.
Two Numbers Example
Numbers: 15, 25
GCF = 5, LCM = 75
Prime Numbers Example
Numbers: 7, 11, 13
GCF = 1, LCM = 1001
Frequently Asked Questions
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