Greatest Common Factor Calculator - Find GCF/GCD
Calculate the greatest common factor (GCF/GCD) of 2-5 numbers using the Euclidean algorithm with detailed prime factorization
GCF Calculator
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What is a Greatest Common Factor Calculator?
A Greatest Common Factor Calculator (also known as GCD or HCF calculator) is a free mathematical tool that finds the largest positive integer that divides two or more numbers without leaving a remainder. This calculator uses the efficient Euclidean algorithm to compute the GCF of 2-5 numbers instantly.
This calculator is perfect for:
- Students - Simplify fractions and solve number theory problems
- Teachers - Demonstrate prime factorization and divisibility concepts
- Mathematics Enthusiasts - Quick verification of GCF calculations
For finding the smallest common multiple of numbers, explore our Least Common Multiple Calculator to calculate LCM using prime factorization.
To find all factors of a single number, use our Factor Calculator to identify all divisors and factor pairs.
For breaking numbers down into prime components, check our Prime Factorization Calculator for complete prime factor decomposition.
To identify all common factors between numbers, visit our Common Factor Calculator for detailed factor analysis.
How the Euclidean Algorithm Works
The calculator uses the Euclidean algorithm, one of the oldest and most efficient methods for finding GCF:
Divide larger number (a) by smaller number (b) to get remainder (r)
Replace a with b, and b with remainder r
Continue until remainder is 0. The last non-zero remainder is the GCF
For 3+ numbers, apply GCF iteratively
Key GCF Concepts
Common Factor
A number that divides two or more numbers evenly without remainder. All numbers share 1 as a common factor.
Prime Factorization
Breaking down a number into its prime number components. GCF is found by multiplying common prime factors.
Coprime Numbers
Numbers with GCF = 1 are called coprime or relatively prime. They share no common factors except 1.
Divisibility
The GCF always divides each of the original numbers evenly. It's the largest such divisor.
How to Use This Calculator
Select Number Count
Choose how many numbers (2-5) you want to find GCF for
Enter Numbers
Input positive integers (whole numbers greater than 0)
Calculate GCF
Click Calculate to see the GCF and detailed steps
View Results
See GCF, prime factorization, and all common factors
Benefits of Using This Calculator
- • Instant Results: Calculate GCF of up to 5 numbers immediately using the efficient Euclidean algorithm.
- • Step-by-Step Process: See detailed calculation steps showing how the Euclidean algorithm works.
- • Prime Factorization: View complete prime factor decomposition of each number.
- • All Common Factors: See the complete list of factors that divide all input numbers.
- • Educational Tool: Perfect for learning number theory, fraction simplification, and divisibility rules.
Important GCF Properties
1. GCF is Always Positive
The GCF is always a positive integer, even if negative numbers are involved (we use their absolute values).
2. GCF ≤ Smallest Number
The GCF cannot be larger than the smallest number in your set. It's always a divisor of all numbers.
3. Relationship with LCM
For two numbers a and b: GCF(a,b) × LCM(a,b) = a × b. This is a fundamental property in number theory.
4. Associative Property
GCF(a,b,c) = GCF(GCF(a,b), c) = GCF(a, GCF(b,c)). The order of calculation doesn't matter.
Frequently Asked Questions (FAQ)
Q: What is the greatest common factor (GCF)?
A: The greatest common factor (GCF), also known as greatest common divisor (GCD), is the largest positive integer that divides each of the numbers without a remainder. For example, the GCF of 12 and 18 is 6.
Q: How does the Euclidean algorithm work?
A: The Euclidean algorithm finds the GCF by repeatedly dividing and taking remainders. For two numbers a and b, divide a by b to get remainder r. Then replace a with b and b with r, repeating until the remainder is 0. The last non-zero remainder is the GCF.
Q: Can I find the GCF of more than two numbers?
A: Yes! This calculator supports 2-5 numbers. For multiple numbers, find GCF(a,b) first, then find GCF of that result with c, and continue. The GCF property is associative: GCF(a,b,c) = GCF(GCF(a,b), c).
Q: What's the difference between GCF and LCM?
A: GCF (Greatest Common Factor) is the largest number that divides all given numbers, while LCM (Least Common Multiple) is the smallest number that is divisible by all given numbers. They are inverse concepts in number theory.