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Jul 10, 2024 | Castle Hills

Find everything you need to get started with factors in math, from simple definitions to easy-to-follow examples, guides to factorization, and fun quizzes.

A factor is a number that divides another number evenly. In other words: The factors of a number all go into that number evenly. For example, take the number 12: the numbers that can divide it evenly (its factors) are 1, 2, 3, 4, 6, and 12.

Why?

Because if we divide 12 by 1, we get 12. No decimal points or fractions, just a nice, “clean” 12.

When we divide 12 by 2, we get 6. No remainders there either.

When we divide 12 by 3, we get 4.

And so on.

Every whole number has at least one factor because every number can be divided by 1. Every number greater than 1 has at least two factors because they can be divided by 1 and the number itself.

For example, the number 2 can be divided by 1, and the result is 2. It can also be divided by itself: 2 ÷ 2 = 1.

The number 3 can be divided by 1 and by itself, too, and so can numbers 4, 5, 6, and every other whole number.

Let’s look at more examples of factors:

Just as division helps us understand (and find) factors in math, so can multiplication. It is the same process, but in reverse: instead of dividing numbers to find their factors, we can multiply numbers to see which number they create. The numbers we multiply are the factors of the number we create.

For example, we can multiply 2 by 3:

2 × 3 = 6

The numbers 2 and 3 are factors of 6.

What are the other factors of 6? Which other numbers can we multiply to create 6?

How about the number 1?

1 × 6 = 6

Yes! The numbers 1 and 6 are also factors of 6.

But, how about the number 4 — can we multiply 4 by another whole number to create 6?

No, we cannot, so the number 4 is not a factor of 6.

Here are some more examples of how we can use multiplication to find factors of other numbers, in this case, the number 24:

Just as division helps us understand (and find) factors in math, so can multiplication. It is the same process, but in reverse: instead of dividing numbers to find their factors, we can multiply numbers to see which number they create. The numbers we multiply are the factors of the number we create.

For example, we can multiply 2 by 3:

2 × 3 = 6

The numbers 2 and 3 are factors of 6.

What are the other factors of 6? Which other numbers can we multiply to create 6?

How about the number 1?

1 × 6 = 6

Yes! The numbers 1 and 6 are also factors of 6.

But, how about the number 4 — can we multiply 4 by another whole number to create 6?

No, we cannot, so the number 4 is not a factor of 6.

Here are some more examples of how we can use multiplication to find factors of other numbers, in this case, the number 24:

**Factor pairs** are simply two numbers that you can multiply together to get a specific number.

For instance, if we take the number 8, its factor pairs are 1 and 8, and 2 and 4, because 1 × 8 = 8 and 2 × 4 = 8.

Sometimes students confuse factors for multiples, so let’s take a quick look at how the two differ.

**Factors **of a number are the whole numbers that can be multiplied together to make that number. For example, if we look at the number 12, its factors are 1, 2, 3, 4, 6, and 12. When we multiply 1 and 12, 2 and 6, or 3 and 4, we get 12.

**Multiples**, on the other hand, are numbers that we get when we multiply one whole number by another. If we take the number 4 and multiply it by 1, we get 4. Then if we multiply 4 by 2, we get 8. And if we continue multiplying the number 4, we’ll get 12, 16, 20, and so forth. These are all multiples of 4 because they’re what we get when we keep multiplying 4 by whole numbers.

So, **factors** are the numbers that multiply together to make a larger number or that can divide that larger number evenly, while **multiples** are the numbers we get when we multiply numbers together.

Let’s see it in action with another example.

For the number 6:

**Factors:** 1, 2, 3, and 6. This is because 1 x 6 = 6, and 2 x 3 = 6.

**Multiples**: To get the multiples of 6, we multiply it by consecutive whole numbers. 6 x 1 = 6, 6 x 2 = 12, 6 x 3 = 18, 6 x 4 = 24, and so on.

Great work so far! Let’s round up our definition of factors by listing some of their key properties:

**Every whole number has 1 as its smallest factor.**This means that every whole number can be divided by 1 without anything left over.**The biggest factor of any number is the number itself.**For example, if we take the number 20, it can be divided by 20, making 20 the biggest factor of itself.**Every number has a limited number of factors.**For example, if we take the number 10, its factors are 1, 2, 5, and 10. There’s a finite number of factors, which means the list eventually ends.**A factor is always smaller or equal to the number it goes into**. For example, if we have the number 12, its factors include 1, 2, 3, 4, 6, and 12.**Apart from 0 and 1, every whole number has at least two factors.**One of them is 1 and the other is the number itself.

Understanding factors is the first step, but now let’s learn how to find them! There are three methods we can use: the multiplication method, the division method, and the rainbow method.

To find factors using multiplication, we look for** factor pairs that, when multiplied**, create the number we’re interested in.

Let’s say we want to find factors of the number 84.

To find factors through multiplication, we look for pairs of whole numbers that multiply to give us the target number. For instance:

- 1 × 84 = 84
- 2 × 42 = 84
- 4 × 21 = 84

So, the factors of 84 we can easily find using multiplication are: 1, 2, 4, 21, 42, and 84

To find factors using division, we **divide the target number** by different numbers, starting with 1 and ending with the number itself, to see if they divide it evenly. Keep in mind that finding factors through division can take time, so it’s best used once you’ve come up with the factor pairs using the multiplication method.

Now, let’s see how division helps us find factors of 84.

We already know that 1 and 2 are factors of 84, but what about 3?

84 ÷ 3 = 28. This means we can add 3 and 28 to the list of factors of 84.

Next, we know for sure that 4 and 21 are factors but are not sure about the numbers that come in between 4 and 21.

What we’ve learned from multiplication tables helps us eliminate some numbers. We can confidently remove 5 and 15 because multiplying anything by those two numbers produces a number ending in either 0 or 5 (e.g., 5 x 9 = 45, 15 x 4 = 60). We can leave out 10 and 20 too as multiplication by those numbers will always give us a number ending in 0.

The next two numbers to test would be 6 and 7.

- 84 ÷ 6 = 14
- 84 ÷ 7 = 12

To recap, here are the factors of 84 we’ve found using the division method so far are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28.

You can keep going like this to find more factors of 84.

The rainbow method helps us organize the factors we’ve already found using the multiplication and division methods. It also enables us to see any other numbers we have left to test. We list out our factors and draw rainbow-shaped arcs to connect factor pairs.

Let’s see how it’s done:

The diagram above shows us factors of 18. Here’s a step-by-step explanation.

- We know 1 and 84 are factors, and 2 and 42 are factors of 84.

There are no other whole numbers between 1 and 2 that could be factors of 84, so there are no other whole numbers between 42 and 84 that could be factors of 84.

- Next, we look at 3 and 28. There are no whole numbers between 2 and 3, so there are no other factors between 28 and 42.
- Then, we look at 4 and 21. Since there are no whole numbers between 3 and 4, so there are no other factors between 21 and 28.
- Now, let’s check 6 and 14. There are no whole numbers between 4 and 6 that are factors of 84.
- After that, we look at 7 and 12. There are no whole numbers between 6 and 7, so there are no other factors between 12 and 14.
- The last thing we want to do is check whether any whole numbers between 7 and 12 are factors of 84.

8,9,10,11 are not factors of 84, so we’ve found them all.

So far, you’ve learned what factors are and how to find them.

When working with advanced 9-year-old students, we like to take a step further and look at the different types of factors. Let’s expand your knowledge of factors:

To understand prime factors, we need to **start with prime numbers**.

A prime number is a special kind of number. Most numbers have lots of factors, like 6 which has 4 factors: 1, 2, 3, and 6.

Prime numbers are different. **They only have two factors:** 1 and the number itself.

For example, 2, 3, 5, and 7 are prime numbers because they can only be divided evenly by 1 and themselves.

On the other hand, **prime factors **are the prime numbers that can multiply together to make a given number.

For example, to find the prime factors of 12, we look for the prime numbers that, when multiplied, equal 12.

- Start with the number 12.
- Check if 2, the smallest prime number, can divide 12 evenly. 12 ÷ 2 = 6.
- Then, continue dividing 6 by 2 because 2 is a factor of 6 as well. 6 ÷ 2 = 3.
- Finally, 3 is a prime number, and it can’t be divided further.
- So, the prime factors of 12 are 2, 2, and 3.

Putting it all together:

2 × 2 × 3 = 12.

Let’s put our knowledge of prime factors into action.

**Prime factorization** is the action of breaking down a number into prime factors, which, if you remember from earlier, are the numbers that themselves have only two factors: 1 and the number itself.

The easiest way to find the prime factors of a number is to start dividing it by prime numbers, such as 2, 3, 5, or 7. Then we keep dividing until we can’t anymore.

**Here’s a step-by-step example:**

Let's do the prime factorization of the number 24.

- We start by finding any factor pair of 24 that includes a prime number. The smallest prime number is 2, so we can start by dividing 24 by 2.

24 ÷ 2 = 12

- Now, we examine 12. Since 12 is still divisible by 2, we continue dividing:

12 ÷ 2 = 6

- Now, we examine 6. We continue dividing by 2:

6 ÷ 2 = 3

- At this point, we’ve reached a prime number (3). Since 3 is prime, we stop.
- Now, let’s gather our prime factors: 2, 2, 2, and 3.

We write the prime factorization of a number by listing its prime factors multiplied together from least to greatest.

In our case, the prime factorization of 24 is: 2 x 2 x 2 x 3.

Let’s try again, but this time with a higher number. We’ll do the prime factorization of 63 together.

- As the smallest prime number (2) doesn’t go into 63, which is an odd number, let’s divide 63 by the next smallest prime number, which is 3:

63 ÷ 3 = 21

- Now, we examine 21. Since 21 is still divisible by 3, we continue dividing:

21 ÷ 3 = 7

- At this point, we’ve reached a prime number (7). Since 7 is prime, we stop.

- Now, let’s gather our prime factors: 3,3,7

The prime factorization of 63 will be: 3 × 3 × 7

A common factor is a number that can divide two or more other numbers without leaving a remainder. In other words, it is a factor two or more numbers share.

Let’s take numbers 4 and 8 and list out all the factors of both numbers:

**Factors of 4: 1, 2, 4****Factors of 8: 1, 2, 4, 8**

Their common factors, i.e., the factors they share, are 1, 2, and 4. All these numbers can divide both 4 and 8 into numbers without remainders.

How about the number 8 — can it be a common factor of numbers 4 and 8?

The answer is no.

In this case, 8 can divide 8 into a whole number (8 ÷ 8 = 1) but cannot divide 4 (4 ÷ 8 = 0.5). So, 8 is not a common factor of 4 and 8.

You can think of a prime factor as the building block of a number, and a common factor as a bridge connecting two numbers together.

The greatest common factor (GCF) is the biggest number that goes into two or more numbers without any remainders. It’s like finding the largest puzzle piece that fits perfectly into two or more numbers you’re looking at.

Let’s take numbers 12 and 18.

To find their GCFs, we can list all the factors of each number.

**Factors of 12 are 1, 2, 3, 4, 6, 12.****Factors of 18 are 1, 2, 3, 6, 9, 18.**

Now, we look for the largest number that appears in both lists. In this case, the largest number that’s common to both lists is 6. Therefore, the greatest common factor of 12 and 18 is 6.

Watch this video to learn more about the greatest common factor.

Find answers to common questions regarding the identification, properties, and applications of factors.

A prime number is a special kind of number that can only be divided evenly by 1 and itself, such as 2, 3, 5, and 7.

Prime factors are the smallest prime numbers that, when multiplied together, make up a given number. For example, to find the prime factors of 20, we look for the smallest prime numbers that, when multiplied, equal 20, like 2 × 2 × 5.

It’s also important to note that a number can be a prime number and a prime factor at the same time. Take the number 5. It’s a prime number because it’s only divisible by 1 and 5. Additionally, it’s also a prime factor of 20 since 20 can be broken down into its prime factors as 2 × 2 × 5.

No, not every number can be a factor of another number. Factors are numbers that divide evenly without any remainders. For example, 5 is not a factor of 12 because it doesn’t divide it evenly.

No, 0 cannot be a factor of any number because when we multiply any number by 0, we always get 0.

Yes, factors are always whole numbers. They can’t be fractions or decimals.

For example, when we talk about the factors of 12, we say 1, 2, 3, 4, 6, and 12, all whole numbers. Factors are like the building blocks of numbers, always whole.

We use factorization in daily life, often without realizing it. We usually use it when we want to divide things up evenly, without a remainder. Let’s say you have 12 cupcakes, and you want to see how many friends can receive an even number of cupcakes. If you want to give cupcakes to 6 friends, each can get 2, or if you want to share them with 4 friends, each can get 3, and so on.

Students learn about factors in elementary school, usually in the 4^{th} grade.

See how Mathnasium’s tutors help 4^{th} grade students master math.

Mathnasium’s specially trained tutors work with students of all skill levels to help them understand and excel in any K-12 math topic, including factors.

Our tutors assess each student’s current skills and considers their unique academic goals to create personalized learning plans that will put them on the best path to math mastery.

Find a Mathnasium Learning Center near you, schedule an assessment, and enroll today!** **

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