Greatest common factor (GCF)

Before you rely on it

What to check

Verify the result divides both original numbers exactly.

The common error

Where people go wrong with greatest common factor (GCF)

Stopping at a common factor instead of the greatest one. The fraction is smaller and still not in lowest terms, and the error survives every subsequent step.

Worked through

Greatest common factor (GCF) in plain numbers

Find the GCF of 48 and 180. Factor them: 48 = 2⁴ × 3 and 180 = 2² × 3² × 5. Take each shared prime to its smallest power: 2² × 3 = 12. That is the largest number dividing both evenly; it simplifies the fraction 48/180: divide top and bottom by 12 to get 4/15, which cannot be reduced further; the Euclidean method gets there faster with big numbers. Divide 180 by 48, remainder 36. Divide 48 by 36, remainder 12. Divide 36 by 12, remainder 0. The last nonzero remainder, 12, is the GCF.

Easily confused

Greatest common factor (GCF) and its neighbours

The GCF is not the least common multiple, though the two are linked; the GCF is the largest number that divides all the numbers. The LCM is the smallest number they all divide into. For two numbers, GCF × LCM equals the product of the numbers, so 12 × 720 = 48 × 180; it is also called the greatest common divisor, the highest common factor and, in some places, the greatest common measure. Coprime numbers have a GCF of 1. Do not confuse it with the prime factors themselves, which are only the building blocks.

In real decisions

Where greatest common factor (GCF) turns up

You use it to reduce fractions and ratios to lowest terms, to split things into equal groups without leftovers and to simplify algebraic expressions; a teacher divides 48 pencils and 180 sheets of paper into identical kits: at most 12 kits, each with 4 pencils and 15 sheets. Tiling problems use it to find the largest square that covers a rectangle exactly. In computing it appears in cryptography and in simplifying rational numbers.

Know the edges

Where greatest common factor (GCF) stops being reliable

It applies to integers. For very large numbers, factoring is slow, and the Euclidean algorithm is the practical route; a GCF of 1 means no simplification is possible, which is not the same as a wrong answer. With zero or negative numbers, conventions differ slightly. When simplifying algebraic expressions, remember the GCF includes variable parts as well as numbers.

Quick answers

What people ask about greatest common factor (GCF)

What is the GCF of two prime numbers? One, unless they are the same prime. Distinct primes share no factor other than 1, so they are coprime.

How do I use the GCF to simplify a ratio? Divide both terms by their GCF. A ratio of 48 to 180 becomes 4 to 15 after dividing by 12. The result cannot be reduced further.

What is the fastest method for large numbers? The Euclidean algorithm. Repeatedly divide, keeping the remainder, until the remainder is zero. The last nonzero remainder is the GCF.