## Table of Contents

- What is Prime Factorization?
- Important Concepts in Prime Factorization:
- Serial Factorization Method of Prime Factorization:
- Factor Tree Method of Prime Factorization:
- Using Prime Factorization:
- Lesson Summary:

What is prime factorization? How about prime factors and a factor tree model? With this lesson you will learn the definition of prime factorization, explanation of prime factors, factor tree model, and how prime numbers are used.
Updated: 06/28/2021

- What is Prime Factorization?
- Important Concepts in Prime Factorization:
- Serial Factorization Method of Prime Factorization:
- Factor Tree Method of Prime Factorization:
- Using Prime Factorization:
- Lesson Summary:

**Prime factorization** can be used as both a noun and a verb. Essentially, it is the process of obtaining the unique prime numbers that multiply into a larger number. The resulting answer is also called the prime factorization. Though the prime factorization definition is simple, the process can look like a tricky topic on the surface; however, it can be broken down into manageable steps.

We have multiple methods for finding a number's prime factorization. Being able to quickly find a number's prime factorization helps with simplifying and combining fractions. Once we know the prime factorizations of two numbers, we can use that to easily find the greatest common factor and least common multiple of those numbers.

Let's begin breaking down prime factorization into manageable steps by looking first at its components: prime numbers. Prime factors go hand in hand with prime numbers.

- A
**prime number**is a number that can only be divided by 1 and itself evenly. - Prime factors, explained in a little more detail, are prime numbers that can be multiplied to create a larger number.

5 is an example of a prime number because it can only be divided by 5 and 1 evenly.

We also have a name for non-prime numbers:

- Composite numbers: a
**composite number**is a positive whole number that is not prime and can be divided into smaller whole numbers.

Each composite number has a special series of prime numbers that can be multiplied to get the larger number. This is what makes the prime factors for a number special; the combination of prime factors is unique to each composite number.

To see what prime factorization means, we can apply it to a composite number. Let's start by going over the prime factors that make up 24.

We can check to see if we have the prime factors of a number by multiplying together that string of numbers. If the result is equal to our original number, then we have the factors. If the factors can only be divided by 1 and themselves then they are all prime factors.

For instance, if we look at the number 24:

- The prime factors would be 2 * 2 * 2 * 3.
- If we multiply that back, then we get 2 * 2 * 2 * 3 = 24.
- This combination of prime factors is unique to the number 24. We could not multiply the same prime factorization to get 26, for example. So, each prime factorization is unique to an individual number.

The prime numbers under 30 are very commonly used, so it may be helpful to memorize them.

It is especially important to know common prime factors when working with the serial factorization method of prime factorization. With **serial factorization**, we factor out each prime number by beginning with the smallest prime number and increasing the size of the prime number until the remaining factor is prime.

Let's look at some prime factorization examples to help determine how to work with the serial factorization method.:

- If we have the number 96 and are looking for the prime factors that go into it we know we can first factor out a 2. It is the smallest prime factor that goes into 96, giving us 2 * 48.
- We can factor out another 2 from 48 as it is the smallest prime factor that goes into 48, giving us 2 * 2 * 24.
- Again, we can factor out a 2, leaving us with 2 * 2 * 2 * 12.
- We can factor out a 2 from 12 and are left with the following: 2 * 2 * 2 * 2 * 6.
- Finally, we can factor out a 2 from 6 and we are left with 2* 2 * 2 * 2 * 2 * 3.
- All the numbers in this number string are prime, so it is our prime factorization for the number 96.

As another example, here are the steps for finding the prime factorization of 126:

- 2 goes into 126, 63 times.
- 3 goes into 63, 21 times.
- 3 goes into 21, 7 times.
- Since 7 is a prime number, that is where we stop.
- The final answer is the prime factorization of 126 = 2 * 3 * 3 * 7.

We can also line prime factors up in a tee chart to help organize the process of serial factorization. We see that the prime factors are circled on the left side of the chart. The final number we end up with on the right side will also be a prime factor. Look at the example with the number 126.

The **factor tree** method is performed just as it sounds, by placing the factors in a tree shaped formation downward from the original number. The difference between the factor tree method and serial factorization is that instead of starting with the smallest prime factor that will go into the number, we can start with any combination that will go into the number. Once a prime factor is found, that branch of the tree comes to a stop.

Let's look at some prime factorization examples to help determine how to work with the factor tree model:

Using the factor tree method with 96, we have the following steps:

- 96 can be broken down into 6 and 16.
- 6 can be broken down further into 2 * 3, which are both prime numbers. The branch of the tree stops on this side.
- The 16 can be broken down into 2 * 8. The 2 is prime so it stops.
- The 8 can be broken down into 2 * 4. The 2 is prime so it stops.
- The 4 can be broken down into 2 * 2.
- This leaves the final prime factorization for 96 as 2 * 2 * 2 * 2 * 2 * 3.

There are a wide variety of uses for prime factorization, from simplifying fractions to modern day cyber encryption, but two of the more common uses of prime factorization are the two we will be working with next: finding the greatest common factor and the least common multiple. These two uses help to bridge the gap between prime factorization as seen in the pure prime factorization examples above and the use of prime numbers in daily life.

Prime factorization and greatest common factors (GCF) go hand in hand. The **greatest common factor** is the number that is the largest number (or factor) that will divide into two different numbers. We can determine the GCF quickly by knowing what prime factors a number has in common. Let's use the numbers that have already been factored: 96 and 126.

- The prime factorization of 96 is 2 * 2 * 2 * 2 * 2 * 3.
- The prime factorization of 126 is 2 * 3 * 3 * 7.

The GCF of 96 and 126 can be found by multiplying the prime factors found in their prime factorizations that they have in common: 2 * 3 = 6. The greatest common factor between 96 and 126 is 6.

Greatest common factors can also used for simplifying fractions. For example, we can simplify 96 over 126 by dividing the top and bottom by its GCF of 6.

The **least common multiple** (LCM) is the smallest number that two numbers can be multiplied into; it can also be described as the lowest common multiple. A multiple of a number is the number we get when we multiply it by another number. An example of this would be the multiples of 3: 3, 6, 9, 12, 15, 18... It is important to be able to find the LCM of two numbers for finding common denominators while adding and subtracting fractions.

To find the LCM between two numbers, we can compare their lists of multiples:

For example, here are the lists used to find the LCM of 2 and 3:

- Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16
- Multiples of 3: 3, 6, 9, 12, 15, 18

The lowest multiple they share is 6, so 6 is the LCM of 2 and 3.

While the list method of finding least common multiples can give us a better understanding of what multiples are, we can use prime factorization to easily determine the LCM of any two numbers.

Let's look at the prime factorization of 21 and 9 to determine the LCM of the two numbers. This process is also shown in the corresponding image.

Finding the LCM of 21 and 9, we start out by looking at the prime factorization for each number.

- The prime factorization of 21 is 7 * 3, and the prime factorization of 9 is 3 * 3.
- Line up the corresponding numbers. The two prime factorizations have one 3 in common.
- To find the LCM, we bring down the numbers to multiply, only bringing down the numbers once if there are several of the same number in one column.
- This leaves us with 7* 3 * 3, which is 63.
- The least common multiple of 21 and 9 is 63.

In this lesson we learned about the following:

**Prime factorization**is both the method in which we obtain the prime numbers that multiply into a larger number and the answer we come up with.- A
**prime number**is a number that can only be divided by 1 and itself evenly. - A
**composite number**is a positive whole number that is not prime, so it can be divided into smaller whole numbers other than itself and 1. **Serial factorization**is a prime factorization method that begins with the smallest prime factor, then increases the size until the largest remaining factor is prime. We can use a tee chart to organize the numbers.- A
**factor tree**finds the prime factorization by placing the factors of a number in a tree-shaped formation and going downward from the original number. Each tree branch terminates on a prime number, and it does not matter which factors we begin with. - The
**greatest common factor**is the number that is the largest number (or factor) that will divide into two different numbers. - The
**least common multiple**is the lowest number that two or more numbers can be multiplied into.

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Frequently Asked Questions

There are a couple methods for finding the prime factorization of a number. These include using a factor tree and serial factorization.

Prime factorization (verb): It is the method in which we obtain the prime numbers that multiply into a larger number.

Prime factorization (noun): It is also what we call the final answer of one of these problems.

The prime factorization of 36 is 2 * 2 * 3 * 3. You can get to this my breaking down 36 into 2 and 18. 2 is prime. 18 can then be broken down into 2 and 9. This 2 is your second 2 in the prime factorization. Then, the 9 can be broken down into 3 and 3. 3 is prime, leaving the final answer as 2 * 2 * 3 * 3

You can find the prime factorization of a number by a few different methods: a factor tree or serial factorization. Using a factor tree, once a component can no longer break down evenly, it is a prime factor. For instance, finding the prime factorization of 12. First, break it into 3 * 4. 3 is prime, but 4 can be broken down further to 2 * 2. Since 2 is prime the prime factorization of 12 is 3 * 2* 2.

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