What is GCF (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 all given numbers evenly without leaving a remainder. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 (12÷6=2) and 18 (18÷6=3) evenly. GCF is fundamental in mathematics for simplifying fractions, solving ratio problems, and understanding number relationships. Unlike LCM which is always ≥ the largest input, GCF is always ≤ the smallest input number.

How do you calculate GCF?

There are three main methods to calculate GCF: (1) Euclidean Algorithm (most efficient for 2 numbers): Repeatedly divide and use remainders until reaching 0. Example: GCF(48,18): 48=18×2+12, 18=12×1+6, 12=6×2+0, so GCF=6. (2) Prime Factorization Method: Break numbers into prime factors and multiply the common ones with lowest powers. Example: 24=2³×3, 36=2²×3². Common: 2²×3=12, so GCF=12. (3) Listing Factors Method: List all factors of each number and find the greatest common one. Example: Factors of 12: 1,2,3,4,6,12. Factors of 16: 1,2,4,8,16. Common: 1,2,4. GCF=4. The Euclidean algorithm is fastest for large numbers, while prime factorization is best for understanding and multiple numbers.

What is the relationship between GCF and LCM?

For any two positive integers a and b, there's a fundamental relationship: GCF(a,b) × LCM(a,b) = a × b. This means if you know one value, you can find the other. Example: For 12 and 18, GCF=6 and LCM=36. Check: 6×36=216, and 12×18=216 ✓. This relationship is useful for verification and quick calculations. If two numbers are coprime (GCF=1), their LCM equals their product. If one divides the other, GCF equals the smaller number and LCM equals the larger. This formula only works for exactly two numbers, not three or more.

How do you use GCF to simplify fractions?

To simplify a fraction, divide both numerator and denominator by their GCF. Steps: (1) Find GCF of numerator and denominator, (2) Divide both by GCF, (3) Result is the simplified fraction. Example: Simplify 24/36. Find GCF(24,36)=12. Divide: 24÷12=2, 36÷12=3. Simplified: 2/3. The fraction is fully simplified when GCF=1 (numerator and denominator are coprime). If GCF>1, the fraction can be further simplified. This is why GCF is essential in fraction arithmetic—before adding or subtracting, simplify using GCF for easier calculations and cleaner answers.

What is the GCF of three or more numbers?

To find GCF of three or more numbers, calculate progressively: find GCF of first two, then find GCF of that result with the third number, continue for all numbers. Example: Find GCF(24, 36, 60). Step 1: GCF(24,36)=12. Step 2: GCF(12,60)=12. Final answer: 12. Alternatively, use prime factorization: factor each number, then multiply common primes with lowest powers. Example: 24=2³×3, 36=2²×3², 60=2²×3×5. Common: 2²×3=12. The progressive method works well with calculators, while prime factorization helps visualize why the GCF is what it is. The GCF of multiple numbers is the largest number that divides all of them evenly.

What are real-world applications of GCF?

GCF has many practical applications: (1) Fraction Simplification: Reduce 18/24 by dividing by GCF(18,24)=6 to get 3/4. (2) Equal Groups: Distribute 24 pencils and 36 erasers equally among students—GCF(24,36)=12, so maximum 12 students with 2 pencils and 3 erasers each. (3) Tile Layouts: Find largest square tile for a 24×36 inch space—GCF=12, use 12×12 inch tiles. (4) Recipe Scaling: Scale down a recipe calling for 8 cups flour and 6 cups sugar—GCF(8,6)=2, divide by 2 for half recipe. (5) Music: Simplify time signatures and rhythm ratios. (6) Modular Arithmetic: Solving Diophantine equations and number theory problems. Understanding GCF helps optimize resources, simplify problems, and find efficient solutions.

GCF Calculator - Greatest Common Factor Calculator with Steps | AICalculator.pro

About GCF Calculator

Our GCF Calculator is a free, comprehensive online tool designed to find the Greatest Common Factor (also known as GCD - Greatest Common Divisor or HCF - Highest Common Factor) of 2 to 10 positive integers instantly. Whether you're simplifying fractions, solving division problems, or learning number theory, our calculator provides accurate results with detailed explanations, Euclidean algorithm visualization, and step-by-step solutions.

Understanding GCF: The Basics

What is GCF?

The Greatest Common Factor (GCF) is the largest positive integer that divides all given numbers evenly without leaving a remainder. It represents the biggest "building block" that all the numbers share.

Example

Find GCF(12, 18):

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
GCF(12, 18) = 6 (the largest common factor)

Key Properties of GCF

Always ≤ smallest input: GCF(a,b) ≤ min(a,b)
Divides all inputs: Every input number is divisible by the GCF
Uniqueness: There is only one GCF for any set of numbers
With 1: GCF(a, 1) = 1 for any positive integer a
With self: GCF(a, a) = a

Methods to Calculate GCF

Method 1: Euclidean Algorithm

The Euclidean Algorithm is the most efficient method for finding GCF of two numbers. It uses repeated division to find the greatest common divisor.

Example: GCF(48, 18)

Step 1: 48 = 18 × 2 + 12

Step 2: 18 = 12 × 1 + 6

Step 3: 12 = 6 × 2 + 0

GCF = 6 (last non-zero remainder)

Method 2: Prime Factorization

Break each number into prime factors, then multiply the common primes with their lowest powers.

Example: GCF(24, 36)

24 = 2³ × 3¹

36 = 2² × 3²

Take lowest powers of common primes:

GCF = 2² × 3¹ = 4 × 3 = 12

GCF(24, 36) = 12

Method 3: Listing Factors

List all factors of each number and identify the greatest common one. Best for small numbers.

GCF and LCM Relationship

For any two positive integers a and b:

GCF(a,b) × LCM(a,b) = a × b

This relationship helps verify calculations and find one value from the other

Practical Applications of GCF

1. Simplifying Fractions

The most common use of GCF is simplifying fractions by dividing both numerator and denominator by their GCF.

Example: Simplify 18/24

Step 1: Find GCF(18, 24) = 6

Step 2: Divide both by 6

18 ÷ 6 = 3

24 ÷ 6 = 4

Result: 18/24 = 3/4

2. Equal Distribution

GCF helps divide items into equal groups. Example: Distribute 24 apples and 36 oranges equally among students—GCF(24,36)=12, so maximum 12 students.

3. Tile and Layout Problems

Find the largest square tile that fits perfectly. Example: For a 24×36 inch space, GCF=12, use 12×12 inch tiles.

Related Calculators

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Additional Resources

For more information about GCF and number theory:

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GCF Results

GCF & LCM Reference

What is GCF?

Common Applications

GCF Properties

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About GCF Calculator

Understanding GCF: The Basics

What is GCF?

Example

Key Properties of GCF

Methods to Calculate GCF

Method 1: Euclidean Algorithm

Example: GCF(48, 18)

Method 2: Prime Factorization

Example: GCF(24, 36)

Method 3: Listing Factors

GCF and LCM Relationship

Practical Applications of GCF

1. Simplifying Fractions

Example: Simplify 18/24

2. Equal Distribution

3. Tile and Layout Problems

Related Calculators

LCM Calculator

Fraction Calculator

Factor Calculator

Additional Resources