GCF & LCM Calculator
Calculate the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of multiple numbers. See prime factorizations and step-by-step solutions.
Formula:GCF × LCM = Product of Numbers
Results
GCF
12
Greatest Common Factor
GCF12
LCM72
Common Factors1, 2, 3, 4, 6, 12
Calculate
Enter Numbers
Prime Factorizations
24=2 × 2 × 2 × 3
36=2 × 2 × 3 × 3
How to Find GCF
Step 1: Find prime factorizations
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
Step 2: Find common prime factors
Common factors: 1, 2, 3, 4, 6, 12
Step 3: Multiply common factors
GCF = 2 × 2 × 3 = 12
GCF(24, 36) = 12
All Factors
Factors of 24:
1, 2, 3, 4, 6, 8, 12, 24
Factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
Highlighted numbers are common factors
Verification
For two numbers: GCF × LCM = Product
12 × 72 = 864 = 24 × 36 ✓
Check GCF divides all numbers:
24 ÷ 12 = 2 ✓
36 ÷ 12 = 3 ✓
Check LCM divisible by all:
72 ÷ 24 = 3 ✓
72 ÷ 36 = 2 ✓
Practice GCF & LCM Problems
Test your understanding with real problems
Practice with 6 problems to test your understanding.
Results
GCF
12
Greatest Common Factor
GCF12
LCM72
Common Factors1, 2, 3, 4, 6, 12
?How to Find the GCF
The Greatest Common Factor (GCF), also called Greatest Common Divisor (GCD), is the largest number that divides evenly into all given numbers. To find the GCF: list prime factors of each number, identify common factors, and multiply them together. For example, GCF(12, 18) = 6, because 6 is the largest number that divides both 12 and 18 evenly.
What is the Greatest Common Factor?
The Greatest Common Factor (GCF), also known as Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest positive integer that divides two or more integers without leaving a remainder. It is fundamental in simplifying fractions, finding equivalent ratios, and solving problems involving divisibility.
Key Facts About GCF
- GCF (Greatest Common Factor) is the largest number that divides all given numbers evenly
- GCF is also called GCD (Greatest Common Divisor) or HCF (Highest Common Factor)
- To find GCF using prime factorization: multiply common prime factors with lowest powers
- GCF of any number and 1 is always 1
- GCF of any number and itself is the number itself
- Relationship: GCF(a,b) x LCM(a,b) = a x b for any two numbers
- Two numbers are coprime (relatively prime) if their GCF is 1
- GCF is used to simplify fractions by dividing numerator and denominator by their GCF
Quick Answer
The Greatest Common Factor (GCF), also called Greatest Common Divisor (GCD), is the largest number that divides evenly into all given numbers. To find the GCF: list prime factors of each number, identify common factors, and multiply them together. For example, GCF(12, 18) = 6, because 6 is the largest number that divides both 12 and 18 evenly.
Frequently Asked Questions
The GCF (also called GCD - Greatest Common Divisor) is the largest number that divides evenly into all given numbers. For example, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 without a remainder.
The LCM is the smallest positive number that is a multiple of all given numbers. For example, the LCM of 4 and 6 is 12 because 12 is the smallest number that both 4 and 6 divide into evenly.
List the prime factorization of each number. Find all prime factors that appear in ALL factorizations. Multiply these common factors together. Example: 12 = 2² × 3, 18 = 2 × 3². Common factors: 2 × 3 = 6.
List the prime factorization of each number. Take each prime factor at its highest power that appears. Multiply them together. Example: 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36.
For any two numbers a and b: GCF(a, b) × LCM(a, b) = a × b. This relationship helps verify calculations. Example: GCF(12, 18) = 6, LCM(12, 18) = 36. Check: 6 × 36 = 216 = 12 × 18 ✓
Last updated: 2025-01-15
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Results
GCF
12
Greatest Common Factor
GCF12
LCM72
Common Factors1, 2, 3, 4, 6, 12