# What is Prime Factorization? - Definition & Examples

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• 0:05 Definition of Prime…
• 1:06 Factor Trees
• 5:01 Divide and Conquer
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Lesson Transcript

Jennie has eight years experience teaching high school math and has a Master's degree in Education Leadership.

In this lesson, you will learn about prime numbers and how to find the prime factorization of a number. By the time we're done, you should be skilled at breaking down numbers into their smallest parts!

## Definition of Prime Factorization

Have you ever seen the show Fear Factor? It required contestants to face a variety of fear inducing stunts in order to win the grand prize of \$50,000. At the end of the show, the host would say to the winner, 'Evidently, fear is not a factor for you!' What exactly does that mean? Well, it means that fear doesn't play a part in their actions and decisions. So, then a 'factor' is something that affects an outcome. In mathematics, factors are the numbers that multiply to create another number.

The prime factorization of a number, then, is all of the prime numbers that multiply to create the original number. It would be pretty difficult to perform prime factorization if we didn't first refresh our memory on prime numbers. With that being said, a prime number is a number that can only be divided by one and itself. Here are a few prime numbers to get you started:

Prime Numbers
2
3
5
7
11
13
17
19
23

They might seem like a random bunch of numbers, but they do have that one very important thing in common; they are only divisible by one and themselves.

## Factor Trees

One way to think about solving for the prime factorization of a number is to think about leaves on a tree. The tree is the given number. As we break it down, we create branches, and when we get to the smallest factors, we see the leaves. The connection to trees isn't an accident. In math, we often use factor trees as a method to perform prime factorization. Let's look at an example using the number 70.

Think about factors that will give us a product of 70. Since 70 is even, we know it is divisible by two: 2 * 35 will give us 70.

Two is prime, but 35 is not, so we have to keep going. What factors will give us a product of 35? Five * 7 = 35, so let's break that down once more.

The numbers we are left with cannot be broken down any further. Now that we only have prime numbers left, we can find the prime factorization of 70 by looking at the leaves of our tree.

The prime factorization of 70 is 2 * 5 * 7. But how do we know this is correct? Here's a hint: you can always check your work. Simply multiply your factors to be sure that they result in your original value. Does 2 * 5 * 7 really equal 70? Well, 2 * 5 = 10, and 10 * 7 = 70. So, yes! Good job!

Just in case you were thinking to yourself, 'What if I chose two different factors for 70 during the first step? Would I still get the same answer?' Let's take a look!

This time, let's try a factor tree using two different factors that give us 70. When we multiply 7 * 10, we still get 70. Seven is a prime number because it is only divisible by itself and one. What about ten? There are several ways to get a product of ten, so we can break it down further. Two times five gives us ten, so let's add it to our tree. Since two and five are both prime numbers, they cannot be broken down any further and we are now finished.

The leaves tell you that the prime factorization of 70 is 2 * 5 * 7, so you can see you would still get the same answer! As my grandpa used to say, 'There's more than one way to skin a cat.' You can be confident that no matter which factors you start with, if you keep going until there are only prime factors left, you will get the right answer. Just remember to check your work. Keep in mind it is proper math etiquette to write your answer with the prime values in order from least to greatest. Depending on how you factored your original number, some rearranging may be needed.

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