Greatest Common Factor Calculator
Calculate the Greatest Common Factor (GCF) of two or more numbers with our easy-to-use calculator.
Understanding the Greatest Common Factor (GCF)
Our Greatest Common Factor Calculator helps you find the largest positive integer that divides each of the given numbers without a remainder. The GCF (also known as GCD or HCF) is a fundamental concept in number theory with applications in fractions, simplification, and many other areas of mathematics.
What is the Greatest Common Factor?
The Greatest Common Factor (GCF) of two or more integers is the largest positive integer that divides each of the given numbers without leaving a remainder. For example, the GCF of 24 and 36 is 12, as 12 is the largest number that divides both 24 and 36 without a remainder.
Methods to Calculate GCF:
Using Euclidean Algorithm:
The Euclidean Algorithm is an efficient method to find the GCF of two numbers by performing successive division with remainder:
- If b > a, swap a and b
- Divide a by b to get quotient q and remainder r
- If r = 0, then GCF = b
- Otherwise, set a = b and b = r, and repeat from step 2
Using Prime Factorization:
Find the prime factorization of each number, then multiply the common prime factors with the minimum power to get the GCF.
For example, to find GCF(24,36):
- 24 = 23 × 3
- 36 = 22 × 32
- Common factors: 22 × 3
- GCF = 22 × 3 = 12
Applications of GCF:
In Mathematics:
- Simplifying fractions to lowest terms
- Solving Diophantine equations
- Finding equivalent fractions
- Polynomial factorization
In Real-World Applications:
- Optimizing resource allocation
- Finding the largest possible unit of measurement
- Cryptography and computer security
- Designing patterns and tilings
How to Use the Calculator:
- Enter two or more positive integers separated by commas
- Choose your preferred calculation method
- Select whether to show the calculation steps
- Click "Calculate GCF" to see the results
Interesting Facts About GCF
- GCF and LCM Relationship: For any two numbers a and b, GCF(a,b) × LCM(a,b) = a × b
- Coprime Numbers: Two numbers are coprime (or relatively prime) if their GCF is 1
- Bézout's Identity: For any two integers a and b, there exist integers x and y such that ax + by = GCF(a,b)
- GCF Properties: GCF(a,b) = GCF(a-b,b) if a > b
- Distributive Property: GCF(a,b,c) = GCF(GCF(a,b),c)