Greatest Common Factor Calculator
Find the GCF of two or more numbers using prime factorization and Euclidean algorithm
Separate numbers with commas or spaces
Greatest Common Factor
GCF / GCD / HCF
Least Common Multiple
LCM
GCF × LCM Relationship
GCF(, ) × LCM(, ) = ×
× =
Prime Factorization
Step-by-Step Solution
Common GCF Examples
| Numbers | GCF | Action |
|---|---|---|
| and |
GCF Properties
Basic Properties
- • GCF(a, 0) = a
- • GCF(a, a) = a
- • GCF(a, b) = GCF(b, a) (commutative)
- • GCF(a, b, c) = GCF(GCF(a, b), c)
Important Relationships
- • GCF(a, b) × LCM(a, b) = a × b
- • GCF(a, b) = GCF(b, a mod b)
- • If GCF(a, b) = 1, a and b are coprime
- • GCF divides any linear combination of a and b
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About Greatest Common Factor Calculator
What is a Greatest Common Factor Calculator?
A Greatest Common Factor (GCF) Calculator helps you find the largest positive integer that divides two or more numbers without leaving a remainder. The GCF is also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).
How to Use This Calculator
- Two Numbers Mode: Enter two numbers to find their GCF
- Multiple Numbers Mode: Enter multiple numbers separated by commas to find the GCF of all
- View Step-by-Step: See the Euclidean algorithm and prime factorization methods
- Compare with LCM: View both GCF and LCM for the given numbers
Understanding GCF
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 integers without leaving a remainder.
Example: GCF(24, 36) = 12 because:
- 24 = 12 × 2
- 36 = 12 × 3
- No larger number divides both 24 and 36 evenly
Methods to Find GCF
Prime Factorization Method
- Find the prime factorization of each number
- Identify common prime factors
- Multiply the common prime factors together
Example: GCF(48, 18)
- 48 = 2⁴ × 3 = 2 × 2 × 2 × 2 × 3
- 18 = 2 × 3² = 2 × 3 × 3
- Common factors: 2 × 3 = 6
- GCF(48, 18) = 6
Euclidean Algorithm
GCF(a, b) = GCF(b, a mod b)
This process is repeated until the remainder is 0. The last non-zero remainder is the GCF.
Example: GCF(48, 18)
- 48 = 2 × 18 + 12 → GCF(48, 18) = GCF(18, 12)
- 18 = 1 × 12 + 6 → GCF(18, 12) = GCF(12, 6)
- 12 = 2 × 6 + 0 → GCF(12, 6) = 6
Result: GCF(48, 18) = 6
GCF Properties
- GCF(a, 0) = a
- GCF(a, a) = a
- GCF(a, b) = GCF(b, a)
- GCF(a, b) × LCM(a, b) = a × b
Frequently Asked Questions
What is the difference between GCF and LCM?
GCF finds the largest number that divides all given numbers, while LCM (Least Common Multiple) finds the smallest number that all given numbers divide into.
How do I find the GCF of more than two numbers?
To find GCF(a, b, c), first find GCF(a, b), then find GCF(result, c). Continue this process for additional numbers.
When is GCF used?
GCF is commonly used for simplifying fractions, solving problems involving ratios, finding common denominators, and many algebraic applications.
Tip: The GCF is fundamental in simplifying fractions to their lowest terms. Divide both numerator and denominator by their GCF to get the simplest form.