Highest Common Multiple Calculator
Find the highest common multiple (also called greatest common factor, GCF, GCD or HCF) of any list of integers, with prime factorization shown.
The "Highest Common Multiple" is more commonly called the Greatest Common Factor (GCF), Greatest Common Divisor (GCD) or Highest Common Factor (HCF). All four terms describe the same thing: the largest integer that divides every number in a set without remainder.
Example
For 12, 18 and 24:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Common factors: 1, 2, 3, 6
- Highest: 6
The Euclidean Algorithm
For two numbers, the fastest method is the Euclidean algorithm, in use since around 300 BCE:
GCD(a, b) = GCD(b, a mod b), until b = 0
GCD(48, 18) = GCD(18, 12) = GCD(12, 6) = GCD(6, 0) = 6. Three steps to the answer.
Practical Uses
- Simplifying fractions: 12/18 divided by 6 = 2/3
- Largest equal groups: dividing 24 apples and 36 oranges into the largest equal baskets uses GCF(24, 36) = 12 baskets, each with 2 apples and 3 oranges
- Tiling problems: the largest square tile that fills a rectangle of any two integer dimensions is GCF(width, height)
Use the Full Calculator
The GCF & LCM Calculator handles every scenario described on this page. For the deeper math and worked examples, read the companion guide.