Prime factorization is a method used to highlight the prime factors of a given number. It visibly lays out the prime factors and their multiplicity for a given number. When evaluated, the prime factorization of a number will return an equivalent value. The reason why prime factorization is important is that the properties of numbers can easily be observed.
There are instances when the prime factorizations of numbers are easier to manipulate, so knowing how to write numbers in this form will come in handy later on. In this article, learn what prime factorization is and how to find the prime factorization of numbers. You’ll be equipped with two methods to write down the prime factorization of numbers.
By the end of this discussion, feel confident when finding the prime factorization of different numbers!
What Is Prime Factorization?
Prime factorization is a method used to break down numbers as a product of their prime factors. This means that when the prime factorization of a number is simplified, it returns the original number. Recall that prime numbers are numbers with exactly two factors: 1 and itself. These prime numbers are :{2, 3, 5, 7, 11, 13, 17, …}, so a number’s prime factorization will be the product of these prime numbers.
|
Number |
Prime Factorization |
|---|---|
|
12 |
2 x 3 x 3 = 2 x 32 |
|
25 |
5 x 5 = 52 |
|
40 |
2 x 2 x 2 x 5= 23 x 5 |
|
60 |
2 x 2 x 3 x 5= 22 x 3 x 5 |
Take a look at the four examples of numbers and their corresponding prime factorization. The numbers, {12, 25, 40, 60}, are all composite numbers – these are numbers that have more than two factors. Through the prime factorization, they can be written down in terms of their prime factors.
- The prime factorization of 12 is 2 x 3 x 3, so 12 has 2 and 3 as prime factors.
- Since 25 is the square of 5, from its prime factorization, it only has 5 as a prime factor.
- The third number, 40, has a prime factorization of 2 x 2 x 2 x 5, so it also has two prime factors: 2 and 5.
- Lastly, 60 has three prime factors: 2,3, and 5.
Notice the second form of the prime factorization? When a number is written as the product of the same prime factor multiple times, writing it as a power of the prime number simplifies the prime factorization’s form. This is why 2 x 2 x 2 x 5= 23 x 5 and 2 x 2 x 3 x 5= 22 x 3 x 5 are equivalent prime factorizations of 40 and 60, respectively.
Before moving on to the two methods used to find the prime factorization of numbers, it’s time to work on some questions as a refresher on prime factors and what prime factorization represents.
Problem 1
The prime factorizations of three numbers are shown below. For each case, what is the value of the composite number?
a. 2 x 5 x 7
Recall that the prime factorization of composite numbers is the equivalent expression of the number written as a product of two or more prime factors. To find the value of the number represented by the prime factorization, 2 x 5 x 7, multiply the three prime factors.
\begin{aligned}2 \times 5 \times 7 &= 10 \times 7\\&= 70\end{aligned}This means that the number represented by 2 x 5 x 7 is equal to 70.
b. 2 x 2 x 3 x 3 x 5
The number represented by the prime factorization, 2 x 2 x 3 x 3 x 5, is a composite number that has 2 and 3 as prime factors twice and 5 as a prime factor once. Similar to the previous problem, evaluate the prime factorization to find the value of the number.
\begin{aligned}2 \times 2 \times 3 \times 3 \times 5 &= 2^2 \times 3^2 \times 5\\&= 36\times 5\\&=180 \end{aligned}Hence, the prime factorization, 2 x 2 x 3 x 3 x 5, represents the composite number, 180.
c. 23 x 3 x 52
The composite number we’re looking for is a number that contains 2 as a prime factor three times, 3 once, and 5 as a prime factor twice. In expanded form, 23 x 3 x 52 is equivalent to 2 x 2 x 2 x 3 x 5 x 5. This is why writing the prime factorization using exponents shortens expressions like this.
Now as with the previous items, evaluate the prime factorization to find its corresponding numerical value.
\begin{aligned}2^3 \times 3 \times 5^2 &= 8\times 3 \times 25\\&= 24\times 25\\&= 600\end{aligned}
This means that 23 x 3 x 52 represents the prime factorization of 600.
Now, it’s time to learn how to write numbers as a product of prime factors. In the next section, learn the two most common methods when finding the prime factorization of numbers.
How To Find the Prime Factorization of a Number?
To find the prime factorization of a number, use either of the two methods: 1) using the factor tree or 2) applying the division method. The goal for each method is to break down the composite numbers as a product of their prime factors. To do so, slowly break down the composite number by dividing a prime factor.
In this section, learn how to apply these two methods to write the prime factorization of any number. Let’s begin by learning how to apply the factor tree method to write a number’s prime factorization.
Finding the Prime Factorization Using the Factor Tree
When finding the prime factorization using the factor tree, break down the composite number as a product of two smaller factors. Work on each factor to further break them down as products of even smaller factors (hopefully, into prime factors). Continue this process until there are no more composite numbers left.
Let’s work on writing the prime factorization of 48 while learning about the key steps of the factor tree method.
Step 1: Write the original number in the center then break it down as a product of its two factors.
Step 2: Use the same process to break down each composite factor. For this example, break down 6 and 8 as products of two smaller factors.
When the factors are all prime, mark these factors then focus on any remaining composite factors. For the case of 48, we now have three prime factors: 2 and 3 from 6 and 2 from 8.
Step 3: Continue the process and make sure there are no composite factors left to break down into smaller factors. With 4 the only composite factor left for 48, write 4 as a product of its prime factors.
This shows the complete factor tree of 48. When asked to write the factor tree of the number apply this method to write the number’s prime factorization.
Step 4: Write down all the prime factors and express the original number as the product of these numbers. From the factor tree of 48, its prime factorization is as shown below.
This means that 48’s prime factorization is equal to 2 x 2 x 2 x 2 x 3 or using the exponents, it’s equal to 24 x 3.
Problem 2
Use what you’re learned about the factor tree method and write the prime factorization of the following numbers.
a. 80
Apply a similar process to break down 80 as a product of smaller factors. 80 is a multiple of 10 so start by breaking down 80 as a product of 8 and 10. Continue until you’re left with prime factors, so here’s the factor tree for 80.
From the factor tree, 80 has 2 as the prime factor with a multiplicity of 4 and has 5 as another prime factor. Hence, the prime factorization of 80 is 2 x 2x 2x 2×5 or 24 x5.
b. 114
Since 114 is even and is equal to 2 x 57, focus on the second factor. The sum of 57’s digits is 12, so this factor is divisible by 3. Break down 57 as a product of 3. With all the additional information, here’s the factor tree for 104.
This means that 114 has a prime factorization of 2 x 3 x 19.
Apply a similar process when working with other numbers’ prime factorization using the factor tree method. For now, it’s time to learn how to apply the division method to find the prime factorization of numbers.
Finding the Prime Factorization Using the Division Method
When finding the prime factorization using the division method, slowly break down the number by dividing the smallest prime factor possible for the number. As with the previous method, learn about the key steps while trying to write 48’s prime factorization using the division method.
Step 1: Begin by writing the given composite number then divide it by the smallest possible prime number. For the case of 48, since it’s even, 2 is going to be its smallest prime factor.
Step 2: Repeat the process and continue using the smallest prime factor of the resulting quotient. Here’s what it’ll be like for 48.
Step 3: Once the resulting quotient returns a prime number, it is time to stop. Take note of all the prime factors plus the last quotient. The number’s prime factorization will be the product of all these prime numbers.
For 48, as expected, its prime factorization is equal to 2 x 2 x 2 x 2 x 3 or 24 x 3. Try out this method and know its steps by heart by working on the next problem as well!
Problem 3
Use what you’re learned about the division method and write the prime factorization of the following numbers.
a. 60
When using the division method, always begin with the smallest prime factor of the number. When working with even numbers, always start with 2. Here’s a breakdown of the factors for 60 using the division method.
This shows that 5 is the largest prime factor of 60. Using these factors, 60 has a prime factorization of 2 x 2 x3 x5 or 22 x3 x5.
b. 135
The number, 135, is a multiple of 3 (add its digits to see if it’s divisible by 3). Since 135 is not even, its smallest prime factor is in fact 3. Begin with that and slowly break down 135 as a product of the prime factors using the division method.
From this, 135’s prime factorization is 3 x 3x 3 x5 or 33 x 5.