Greatest Common Factor

Find the Greatest Common Factor (GCF) of two numbers

"Greatest Common Factor" Calculator

Greatest Common Factor of
and

GCF Calculator - Greatest Common Factor of Two Numbers

What is the Greatest Common Factor (GCF)?

The Greatest Common Factor (GCF) of two numbers is the largest positive integer that divides both numbers evenly with no remainder. GCF is also known as Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

How to find the GCF of two numbers?

There are several methods to find the greatest common factor:

  • Euclidean Algorithm — most efficient method for large numbers
  • Prime Factorization — breaking numbers into prime factors
  • Listing Factors Method — suitable for smaller numbers
  • Online GCF Calculator — quick and accurate results

GCF Calculation Examples

Let's look at several examples of finding the greatest common factor:

  • GCF(12, 16): Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 16: 1, 2, 4, 8, 16. GCF = 4
  • GCF(18, 24): 18 = 2 × 3², 24 = 2³ × 3. GCF = 2 × 3 = 6
  • GCF(15, 25): Common factors: 1, 5. GCF = 5
  • GCF(7, 11): Prime numbers, GCF = 1

Other popular GCF calculations: GCF(12, 18), GCF(8, 12), GCF(16, 24), GCF(20, 30), GCF(24, 36), GCF(30, 45), GCF(36, 48), GCF(40, 60)

Applications of GCF in Mathematics

The greatest common factor is widely used in various mathematical areas:

  • Simplifying Fractions — reducing to lowest terms
  • Solving Diophantine Equations — in number theory
  • Cryptography — in encryption algorithms
  • Programming — algorithm optimization
  • Geometry — constructing regular polygons

Properties of Greatest Common Factor

GCF has the following important properties:

  • GCF(a, b) = GCF(b, a) — commutative property
  • GCF(a, 0) = a — for any number a
  • GCF(a, b) × LCM(a, b) = a × b — relationship with LCM
  • If GCF(a, b) = 1, the numbers are called relatively prime

Practical Problems with GCF

The GCF calculator helps solve many practical problems:

  • Dividing objects into equal groups without remainder
  • Determining the largest tile size for flooring
  • Finding common periods of repeating events
  • Simplifying mathematical expressions and fractions

See Also

About "Greatest Common Factor" Calculator

This calculator will help you find the greatest common factor (GCF) of two numbers. For example, it can help you find out what is the Greatest Common Divisor of 12 and 16? (The answer is: 4). Select the first number (e.g. '12') and the second number (e.g. '16'). After that hit the 'Calculate' button.
Greatest Common Factor (GCF) also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) - it is the largest positive integer that divides each of the integers with zero remainder.

FAQ

What is the greatest common factor of two numbers?

It is the largest positive integer that divides both numbers with no remainder. The same value is also called the greatest common divisor (GCD) and the highest common factor (HCF) — three names for one number. For 12 and 16 it is 4.

How do you find the GCF by prime factorization?

Split both numbers into prime factors and multiply the primes they share, each taken as many times as it appears in both factorizations. 12 = 2 × 2 × 3 and 16 = 2 × 2 × 2 × 2 share 2 × 2, so the GCF is 4.

What is the Euclidean algorithm for the GCF?

Euclid described it in the Elements around 300 BC: divide the larger number by the smaller one, then replace the pair with the smaller number and the remainder, until the remainder is 0. For 16 and 12 that is 16 mod 12 = 4 and then 12 mod 4 = 0, so the GCF is 4. It never factorizes anything, which is why it stays fast on numbers far too large to factor.