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.

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.Let’s do another example.

What is the prime factorization of 92 expressed in exponential form? First things first, let’s do a factor tree for 92. We already know that two is a prime number, so let’s try it first. Is there a number we can multiply times two to give us 92? Yes. The number 46! Since 46 is divisible by numbers other than itself and one, such as two, four and so on, we need to keep going. 2 * 23 = 46. Since both of these numbers are prime, we’ve completed our prime factorization.

At this point, we know that the prime factorization of 92 is 2 * 2 * 23. Let’s check our work. 2 * 2 = 4, and 4 * 23 = 92.

However, we aren’t done yet because the question asked us to write our answer in exponential form. Remember that **exponents** are used to indicate the number of times a number is multiplied by itself.

In this case, instead of having 2 * 2, we have to express that as two to the power of two, or two squared. The final answer would look like this:

## Divide and Conquer

This has been interesting and all, but when are you going to use this information? Well, I’m sure that fractions are one of your favorite topics, so let’s touch on those for a moment. What if you had the fraction of 70/92, and you were asked to simplify it? You could use the prime factorization of each number and cancel the common terms.

The common factor of 2 was cancelled from the top and the bottom leaving the simplified fraction of 35/46. You can be confident that your answer is fully simplified because you did the prime factorization of 70 and 92, and there are no other common factors.

This will become a handy part of your math skills as you move into higher-level algebra or when you are working with numbers that you are not very familiar with. You can always rely on prime numbers to simplify things!

## Lesson Summary

Finding the **prime factorization** of a number is really just breaking it down into the smallest numbers possible. Remember that your answer should only contain **prime numbers** or those numbers that are only divisible by themselves and one. In order to check your work, all you need to do is multiply together the factors you found for your answer.

If, when multiplied, the product is the original number you began with, you have successfully completed prime factorization.

## Vocabulary ; Definitions

**Factors** are the numbers that multiply to create another number.**Prime factorization** is all of the prime numbers that multiply to create the original number.**Prime number** is a number that can only be divided by one and itself.**Factor trees** is a method to perform prime factorization.**Exponents** are used to indicate the number of times a number is multiplied by itself.

## Learning Outcome

After viewing this lesson, you should be able to identify prime numbers using prime factorization.