The greatest common factor (GCF) is a term used to describe the biggest number that can divide evenly into two or more numbers. Sometimes, this is also referred to as the greatest common divisor (GCD) or highest common factor (HCF). A factor is a smaller number that can divide evenly into that number. Let’s look at the greatest common factor definition.
Greatest Common Factor Definition
The greatest common factor, then, can be found when you list out the factors of each number and then find the greatest one that is in common in between the two. Let's take a quick example: finding the greatest common factor between 9 and 12. Let's start by factoring. Factoring is the process of finding the factors of a number.
The factors of 9 are 1, 3, and 9 because you can divide 9 evenly by these numbers.
The factors of 12 are 1, 2, 3, 4, 6, 12 because you can divide 12 evenly by these numbers.
Which is the largest number in common between the two lists? We can see that 3 is both a factor of 9 and 12. It is the largest factor they have in common, so we can name 3 as the GCF or greatest common factor.
Least Common Multiples
Another common term that is sometimes paired with GCF is the least common multiple (LCM). A multiple is a larger number the number can be divided into without leaving a remainder. For example, the multiples of 5 are 5, 10, 15, 20, 25, and so on.
The LCM, then, is the smallest number in common when you list the multiples of two or more numbers. What is the least common multiple between 4 and 5? Let's start by listing out the multiples of the given numbers.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, and so on.
The multiple of 5 are 5, 10, 15, 20, 25, 30, 35, 40, and so on.
Which is the smallest number in common between the two lists? We see that 20 is both a multiple of 4 and 5. So is 40. 20 is the smaller number, and so we can call 20 the lcm or least common multiple.
Using the Greatest Common Factor
One usage for the greatest common factor definition is reducing fractions. In order to see if you can simplify the fraction, you need to find the greatest common factor between both the numerator and denominator. If the GCF or greatest common factor is bigger than 1, then the fraction can be reduced.
Let's take the fractions and . The greatest common factor of 3 and 5 is 1 (they are both prime numbers). So, cannot be simplified. The greatest common factor of 4 and 6 is 2. This means that we can reduce the fraction by dividing both the numerator and denominator by 2. As a result, can be reduced to .
To review, the greatest common factor is the term given to the largest factor that is shared between two numbers. Look for the abbreviations GCF (greatest common factor), HCF (highest common factor), or GCD (greatest common divisor).