How the Common Factor Calculator works
This tool finds the greatest common factor (GCF), the least common multiple (LCM), and every factor shared by two or three whole numbers. It uses the Euclidean algorithm, the standard method for computing a greatest common factor without listing and comparing every divisor by hand.
What the calculator does step by step
- Repeatedly divides the larger number by the smaller and keeps the remainder, until the remainder reaches zero — the last nonzero remainder is the GCF (this is the Euclidean algorithm).
- If a third number is entered, applies the same process again: GCF(a, b, c) = GCF(GCF(a, b), c).
- Derives the least common multiple from the GCF using LCM(a, b) = (a × b) ÷ GCF(a, b), then folds in the third number the same way.
- Lists every common factor by finding all divisors of the GCF — a number divides every input evenly if and only if it divides their GCF.
Understanding the output
- GCF (greatest common factor): also called the greatest common divisor (GCD) — the largest whole number that divides all your inputs with no remainder. Used for simplifying fractions to lowest terms.
- LCM (least common multiple): the smallest whole number that all your inputs divide into evenly. Used for finding a common denominator or for figuring out when repeating events line up (events every 4 and 6 days both fall on day 12).
- Coprime numbers: if the GCF is 1, the numbers share no common factor besides 1 — they are called coprime or relatively prime, even though each may have many factors on its own.
Common uses
Finding common factors is a building block for simplifying fractions, reducing ratios, factoring algebraic expressions, and scheduling problems where events repeat on different cycles. It is one of the first number-theory skills taught alongside multiplication and division.
Frequently Asked Questions
What is the difference between GCF and LCM?
The GCF (greatest common factor) is the largest whole number that divides all your inputs evenly, while the LCM (least common multiple) is the smallest whole number that all your inputs divide into evenly. For two numbers a and b, they are related by GCF(a,b) × LCM(a,b) = a × b.
How does the Euclidean algorithm find the GCF?
It repeatedly replaces the larger number with the remainder of dividing it by the smaller number, until the remainder is zero. The last nonzero remainder is the GCF. For example, GCF(24, 36): 36 ÷ 24 leaves remainder 12; 24 ÷ 12 leaves remainder 0; so the GCF is 12.
What does it mean if the GCF is 1?
A GCF of 1 means the numbers are coprime (relatively prime) — they share no common factor other than 1. Coprime numbers can still each have many factors individually; they just don't overlap.
Can I find the common factors of more than two numbers?
Yes. Enter a third number and the calculator finds GCF(a,b,c) by first computing GCF(a,b), then finding the GCF of that result with c. The common factors are always the divisors of the final GCF.