Greatest Common Factor

Welcome to the greatest common factor, or gcf in short. Here we tell you everything about this mathematical concept, aka highest common factor. In particular, we answer what is a greatest common factor, how to calculate it using three different methods, and what it is used for, along with examples.

What is the Greatest Common Factor

If you have been wondering what is the greatest common factor, then you have come to the right site. By reading on you will learn everything about it. If you just want to calculate the greatest common factor of two or more integers, then head straight to our highest common factor calculator below.

Greatest Common Factor Definition

The definition of greatest common factor is: The gcf of two integers a and b is the largest positive integer that divides the numbers a and b without a remainder. Spelled out, it is the greatest common factor of a and b, also known as greatest common divisor (gcd) of a and b. The gcf of two numbers a and b is usually written as gcf(a,b).

Greatest Common Factor Examples

For example, the greatest common factor of a = 15 and b = 45 is 15. Fifteen is the biggest integer which can divide both, 15 as well as 45, without a remainder, that is mod = 0.

If a = 16 and b = 52, then the result equals 4. 16/4 = 4 and 52/4 = 13. There is no bigger number than four for which the Euclidean division of 16 and 52 produces 0 as rest.

How to Find the Greatest Common Factor

Obviously, how to find the greatest common factor is on the forefront of any questions you might have. We are going to show you three methods: intersection, least common multiple and prime factorization. But first we have to start by answering what is a common factor?

What is a Common Factor

Assumed the factors of a are n, p, q, s, u and the factors of b are o, p, q, r, v. The factors intersecting the two sets above are p and q. So p and q are common factors.

Find the Greatest Common Factor by Intersection

Using the intersection factors of a ∩ factors of b from above we already know what are common factors and which are the common factors of our set. To find the greatest common factor all we have to do is identifying the biggest element in the list {p,q}:

If p > q then p is the gcf of a and b. If q > p then q is the gcf(a,b).

For example, with a = 15 and b = 45: The factors of 15 = 1, 3, 5, 15 and the factors of 55 are 1, 3, 5, 9, 15, 45. The common factors of 15 and 45 are 1, 3, 5, 15, because they appear in both sets.

15 is the greatest common factor which appears in both sets.

You can find the factors of any number using our calculator:

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By definition, the negative numbers (rarely used) you see in the output are factors, too.

Find the Greatest Common Factor by LCM

The greatest common factor of a and b can be computed by using the least common multiple aka lcm of a and b. This is the fastest approach if you happen to know the lcm concept:

gcf (a,b) = $\frac{a \times b}{lcm(a,b)}$

gcf (15,45) = $\frac{15 \times 45}{lcm(15,45)} = \frac{675}{45}$ = 15

Find the Greatest Common Factor by Prime Factorization

The gcd of a and b can be found using the prime factorization of a and b:

Be this the prime factorization of a: p1 x p2
Be this the prime factorization of b: p1 x p1 x p2 x p3

The prime factors and multiplicities a and b have in common are: p1 x p2:
p1 x p2 is the gcd of a and b.

For example:

The prime factorization of a is: 3 x 5
The prime factorization of b is: 3 x 3 x 5

The prime factors and multiplicities 15 and 45 have in common are: 3 x 5.
3 x 5 = 15 is the gcf(15/45).

You can find the the prime factorization for any number using this calculator:

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By reading so far you know how to find the gcd of any two integers a and b.

Greatest Common Factor Calculator

Our greatest common factor calculator can compute the gcf of two or more integers. Just enter your comma-separated numbers, this tool then takes care of the rest.

The gcf is...

Bookmark our calculator now as greatest common divisor calculator or highest common factor calculator.

What is the Greatest Common Factor Used for?

Answer: It is helpful for reducing fractions like a / b. Just divide the nominator as well as the denominator by the gcf(a,b) to reduce the fraction to lowest terms.

$\frac{a}{b} = \frac{\frac{a}{gcf(a,b)}}{\frac{b}{gcf(a,b)}} = \frac{a}{b}\hspace{3px}$ in lowest terms.

With b = 45 and a = 15:

$\frac{15}{45} = \frac{\frac{15}{15}}{\frac{45}{15}} = \frac{1}{3}$.

Remember the greatest common factor as the greatest thing for simplifying fractions.

Properties of GCF

The most important properties of the gcf(a,b) are:

  • Commutative property: gcf(a,b) = gcf(b,a)
  • Associative property: gcf(a,b,c) = gcf(gcf(a,b),c) $\hspace{10px}n\hspace{3px}\epsilon\hspace{3px}\mathbb{Z}$

The associativity is particularly useful to get the gcf of three or more numbers; our calculator makes use of it.

Unless you study math, the commutativity and associativity are enough in school, yet the complete list of properties can be found in the referenced site at the end of this article.

Greatest Common Factor
If you have been searching for gcf learning or how to find gcf then you have come to the correct page, too. The same is the true if you typed what is gcf or what is greatest common factor in your favorite search engine.

Note that you can find the greatest common divisor of many integer pairs by using the search form in the sidebar of this website.

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