# Greatest Common Factor

## Free Greatest Common Factor Fact Sheet

Use the greatest common factor fact sheet below to help you get a better understanding of how factors work. You are encouraged to print or download the greatest common factor fact sheet with the PDF link at the bottom of the page.

Download Greatest Common Factor PDF

## GCF and LCM

Hi, and welcome to this video covering the least common multiple and the greatest common factor.

As you know, there are times when we have to algebraically “adjust” how a number or an equation appears in order to proceed with our math work. We can use the greatest common factor and the least common multiple to do this. The greatest common factor is the largest number that is a factor of two or more numbers, and the least common multiple is the smallest number that is a multiple of two or more numbers.

To see how these concepts are useful, let’s look at adding fractions. Before we can add fractions, we have to make sure the denominators are the same by creating an equivalent fraction:

Example for screen: \(\frac{2}{3}+\frac{1}{6} \rightarrow \frac{2}{3} \times \frac{2}{2}+\frac{1}{6} \rightarrow \frac{4}{6} +\frac{1}{6}=\frac{5}{6}\)

In this example, the least common multiple of 3 and 6 must be determined. In other words, “what is the smallest number that both 3 and 6 can divide into evenly?” With a little thought, we realize that 6 is the least common multiple, because 6 divided by 3 is 2 and 6 divided by 6 is 1. The fraction, \(\frac{2}{3}\),is then adjusted to the equivalent fraction, \(\frac{4}{6}\), by multiplying both the numerator and denominator by 2. Now the two fractions with common denominators can be added for a final value of \(\frac{5}{6}\). In the context of adding or subtracting fractions, the least common multiple is referred to as the least common denominator.

In general, you need to determine a number **larger than or equal to** two or more numbers to find their least common multiple.

It is important to note that there is more than one way to determine the least common multiple. One way is to simply list all the multiples of the values in question and select the smallest shared value, as seen here:

This illustrates that the least common multiple of 8, 4, and 6 is 24 because it is the **smallest** number that 8, 4 and 6 can all divide into evenly.

Another common method involves the **prime factorization** of each value. Remember, a prime number is only divisible by 1 and itself.

Once the prime factors are determined, list the shared factors once, and then multiply them by the other remaining prime factors. The result is the least common multiple:

The least common multiple can also be found by **common (or repeated) division**. This method is sometimes considered faster and more efficient than listing multiples and finding prime factors. Here is an example of finding the least common multiple of 3, 6, and 9 using this method:

Divide the numbers by the factors of any of the three numbers. 6 has a factor of 2, so let’s use 2. Nine and 3 cannot be divided by 2, so we’ll just rewrite 9 and 3 here. Repeat this process until all of the numbers are reduced to 1. Then, multiply all of the factors together to get the least common multiple.

Now that methods for finding least common multiples have been introduced, we will need to change our mindset to finding the greatest common factor of two or more numbers. We will be identifying a value **smaller than or equal to** the numbers being considered. In other words, ask yourself, “what is the largest value that divides both of these numbers?” Understanding this concept is essential for dividing and factoring polynomials.

Prime factorization can also be used to determine the greatest common factor. However, rather than multiplying all the prime factors like we did for the least common multiple, we will multiply only the prime factors that the numbers share. The resulting product is the greatest common factor.

Let’s wrap up with a couple of true or false review questions:

1) The least common multiple of 45 and 60 is 15.

The answer is false. The greatest common factor of 45 and 60 is 15, but the least common multiple is 180.

2) The least common multiple is a number **greater than or equal to** the numbers being considered.

The answer is true. The least common multiple is greater than or equal to the numbers being considered, while the greatest common factor is equal to or less than the numbers being considered.

Thanks for watching, and happy studying!