Prime Factorization
Simple How-To w/ 11+ Examples!

Did you know that a factor tree is the easiest way to find the prime factorization of a number?

It’s true!

Let’s jump in and see how this works.

So prime factorization looks at two things:

1. Primes
2. Factors

Recall that factors are numbers we multiply together to get another number.

Well, prime factorization is all about finding which “prime” numbers multiply together to produce the original number.

What Is A Prime Number?

When only two factors of a number are one and itself, then the number is a prime number.

For example, here is a list of some common prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, etc.

What Is A Composite Number?

Now a composite number is a number that has other factors besides one and itself. In essence, composite numbers are numbers other than prime numbers.

But it is important to note that 0 and 1 are not considered prime or composite.

Factor Tree (How-To)

Okay, so now that we remember what prime numbers are, let’s look at the process of finding those prime numbers that multiply together to get the original number. The method we use is called a factor tree as discussed by Math is Fun.

A factor tree is a diagram with branches that we use to go through the factoring steps. The trick to using a tree diagram is to keep factoring until all the dangling “fruits” are prime numbers.

Example #1 — Factors of 18

Let’s look at an example.

Find the prime factorization of 18.

Notice that we only stopped when all of our dangling fruit were prime numbers.

But what would happen if we started with different factors? Does this change the the problem?

Nope!

Let’s prove it.

Suppose instead of starting with 2 and 9 — we started with 3 and 6.

See, we arrived at the same answer as before, which means no matter which two factors you start with, you will always arrive at the same prime factors!

Notice that all we do to find our prime factors is to divide our given value by a number that goes in evenly. And the product of all of our prime factors will equal the original value. For example,

Example #2 — Integer Factorization of 96

Let’s look at another example.

Find the prime factorization of 96.

Example #3 — Prime Factorization in Reverse

We will also learn how to take the prime factorization and use it to find the original number by applying the order of operations!

For example, find the original number given the following prime factorization.

The Why

But why do we need to find the prime factorization of a number?

What’s the point?

Well, it’s is used when we need to factor polynomials (something we will learn about in Algebra), and it is utilized heavily in code breaking and computer encryption.

Cool!

But for a more practical reason, it’s because we use it when simplifying and combining fractions!

Worksheet (PDF) — Hands on Practice

Put that pencil to paper in these easy to follow worksheets. The handouts are designed so that it doesn’t matter if you’re in 5th, 6th, 7th grade, or beyond — you’ll have everything you need to test your knowledge and learn this material.

Prime Factorization — Practice Problems
Prime Factorization — Step-by-Step Solutions

Video Tutorial — Full Lesson w/ Detailed Examples

You’ll delve into 12 additional explanations and problems with increasing difficulty to enhance your understanding of these techniques. Let’s go!

36 min

• Introduction to Video
• 00:00:31 – What are factors? Prime and Composite Numbers? and How to use a factor tree?

• 00:12:14 – Find the integer factorization for each number (Examples #1-6)
• 00:22:52 – Find the prime factorization (Examples #7-10)
• 00:32:23 – Use reverse factorization to identify the original number (Examples #11-12)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions