In math, the greatest common factor (GCF) is the biggest number that can divide evenly into two or more numbers. People have another name for GCF too, and you may have heard it: Some people call the GCD the highest common factor (HCF). It’s the same thing. A factor is a smaller number that can divide evenly into that number. 

The least common multiple (LCM) is the smallest number in common when you list the multiples of two or more numbers. A multiple is a larger number that the number can be divided into without leaving behind a remainder. 

Now that we have that out of the way, let’s do some math.

 

How to Find the Greatest Common Factor

 

Finding the Greatest Common Factor is not as hard as finding the greatest item on a long restaurant menu, so don’t worry. 

You can find the greatest common factor by making a list of all the factors of each number. Then you look at all the numbers in your list and find the biggest whole number that is in common in between the two. 

Let’s find the greatest common factor between 6 and 12.

 

The factors of 6 are:

1, 2, 3, 6

Why? Because you can divide 6 evenly by these numbers. 

 

Now let’s list the factors of 12:

1, 2, 3, 4, 6, 12

Why? Because you can divide 12 evenly by these numbers. 

 

Now let’s look at both of your lists for 6 and 12. Which is the largest number in common between the two lists? Out of all those numbers, the number that was the greatest between them is the number 6, so that makes 6 the GCF, or the greatest common factor.

Remember that the greatest common factor must be a whole number. No decimals or fractions here!

 

When do We Use the GCF? 

Have you ever needed to reduce fractions? That’s when you will need to know the greatest common factor!  When you need to simplify or reduce a fraction you have to find the greatest common factor between both the numerator and denominator. If the greatest common factor is greater than 1, the fraction can be reduced or simplified.

 

Can you simplify these fractions?

 

7/11 and 4/8.

 

Let’s look at 7/11. Neither 7 nor 11 have many factors as they are prime numbers.

We can multiply 7 by 1 to get 7. And we can multiply 11 by 1 to get 11. That’s it. The greatest common factor between 7 and 11 is the number 1.

That means 7/11 can’t be reduced. 

 

Now let’s look at 4/8.

The factors of 4 are 1, 2, and 4.

The factors of 8 are 1, 2, 4, and 8.

As we can see, the greatest common factor of 4 and 8 is the number 4.

Cool! This means we can reduce the fraction by dividing both the numerator and denominator by 4. 

So, 4/8 can be simplified to 1/2.

 

Got it? Right on! 

Finding the greatest common factor is a great building block for factoring polynomial expressions too!

Least Common Multiples

 

Another term you might often hear paired with GCF is the least common multiple (LCM). Finding the least common multiple isn’t hard either. The least common multiple is the smallest number in common when you list the multiples of two or more numbers. A multiple is a larger number that the number can be divided into without leaving behind a remainder. 

 

Example:

List the multiples of 4:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, and so on counting by 4s…

The least common multiple is the smallest number in common when you list the multiples of two or more numbers. 

 

Exercise:

Can you practice finding the least common multiple between 4 and 6? 

First, list the multiples: 

The multiples of 4:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, and so on counting by 4s…

The multiples of 6:

6, 12, 18, 24, 30, 36, 42, 48, and so on counting by 6s… 

Can you find the smallest number in common between the two lists you made?

That’s right, 12 is is the least common multiple of 4 and 6.

 

Words to know:

Factor: a smaller number that can divide evenly into that number

Greatest Common Factor: The biggest number that can divide evenly into two or more numbers

Highest Common Factor: Another fancy name for greatest common factor

Multiple: A larger number the number can be divided into without leaving a remainder

Least Common Multiple: The smallest number in common when you list the multiples of two or more numbers. 

 

Review: 

The greatest common factor is the largest factor that is shared between two numbers.

The lowest common multiple is the smallest number in common shared between two numbers.