GCF Calculator
GCF6
The greatest common factor (GCF), also called the greatest common divisor (GCD), is the largest positive integer that divides every number in a set without leaving a remainder. This calculator finds the GCF of two or more whole numbers at once using the Euclidean algorithm. It is the key tool for reducing fractions to lowest terms, splitting quantities into equal groups, and simplifying ratios.
Formula
gcd(a, b) = gcd(b, a mod b) until b = 0; GCF(list) folds this across all numbers
- a, b
- A pair of positive integers being compared
- a mod b
- The remainder when a is divided by b
- GCF
- The largest integer dividing every number in the set
How it works
- Enter two or more positive whole numbers. The calculator requires at least two values, and every entry must be a positive integer.
- It computes the GCF by repeatedly applying the Euclidean algorithm: the GCF of the running result and the next number is taken until all numbers have been folded in.
- The single largest factor common to all your numbers is returned. If you enter fewer than two valid integers, no result is produced.
Worked example
Find the GCF of 48, 60, and 72.
- GCF of 48 and 60: 60 mod 48 = 12, 48 mod 12 = 0, so gcd = 12.
- Now combine with 72: 72 mod 12 = 0, so the gcd stays 12.
- No larger integer divides all three numbers.
The greatest common factor of 48, 60, and 72 is 12.
Frequently asked questions
- What is the difference between GCF, GCD, and the common factor calculator?
- GCF and GCD are two names for the same thing: the single largest divisor shared by the numbers. A common factor calculator lists every shared factor, whereas this tool returns only the greatest one.
- Can I find the GCF of more than two numbers?
- Yes. Enter as many positive integers as you like and the calculator folds the Euclidean algorithm across all of them, returning the single largest factor common to the entire set.
- Why must the numbers be positive integers?
- The greatest common factor is defined for positive whole numbers. Decimals and zero or negative values are not accepted because the divisibility relationship the GCF relies on only applies to positive integers.
- How does the GCF help simplify fractions?
- Dividing both the numerator and denominator of a fraction by their GCF reduces it to lowest terms in a single step. For example, the GCF of 12 and 18 is 6, so 12/18 simplifies to 2/3.