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Prime Factorizations
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
?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
What is the GCF (Greatest Common Factor)?
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.
What is the LCM (Least Common Multiple)?
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.
How do I find the GCF using prime factorization?
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.
How do I find the LCM using prime factorization?
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.
What is the relationship between GCF and LCM?
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: 2026-01-15
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
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