topic badge

Extension: Prime factorizations

Introduction

All whole numbers except for 1 are either prime or composite, and composite numbers can always be written as a product of primes. Finding this product (called a prime factoring) can be very useful.

A number is prime if it has exactly two factors: 1, and itself. A number is composite if it has more than two factors.

Factor trees

One of the best ways to find a prime factoring is by using a factor tree. We start with the number we want to investigate, draw a box around it, and draw two lines coming out of it.

A box with 12 inside and two lines coming out of it.

Here is how we might start with the number 12.

A box with 12 inside with two lines coming out connecting to 4 and 3

We then put two numbers that multiply to make 12, such as 3 and 4, at the end of each of the lines.

A 12 in a box connected to 4 in a box and 3 in a circle.

Because 3 is a prime number, we circle it. Since 4 is not a prime number, we draw a box around it instead.

A 12 in a box connected to 4 in a box and 3 in a circle. And 4 is connected to 2, and 2

We then repeat the process with 4, which is 2 \times 2.

A 12 in a box connected to 4 in a box and 3 in a circle. And 4 is connected to 2, and 2 both in circles

And since 2 is prime, we circle both of these numbers.

This is a completed factor tree for 12, and it tells us that 12 = 2 \times 2 \times 3. Multiplying the circled numbers at the end of each branch together always makes the original number.

This image shows a 12 (inside a box) connected to 6 (inside a box) with two lines coming out labeled with a circled 2 and a circled 3,and 2 (inside a circle).

Factor trees are not always unique - here is another factor tree for 12.

Even though the number in the box is different, the numbers at the end of the branches will always be the same for any number - they will just be in a different order.

This image shows a factor tree. Ask your teacher for more information.

Here is a factor tree for 360:

We can therefore write:

\displaystyle 360\displaystyle =\displaystyle 3 \times 5 \times 2 \times 2 \times 2 \times 3
\displaystyle 360\displaystyle =\displaystyle 2 \times 2 \times 2 \times 3 \times 3 \times 5We usually rewrite the factors in ascending order
\displaystyle 360\displaystyle =\displaystyle 2^{3} \times 3^{2} \times 5We can use exponent notation to make the expression shorter

Notice that the factor tree for 12 we made earlier is a smaller part of the factor tree for 360. This is because 12 is a factor of 360, and when we write 360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 we can recognize the prime factoring of 12 inside it: 360 = 2 \times (2 \times 2 \times 3) \times 3 \times 5.

Examples

Example 1

A number has the following factor tree:

The image shows a factor tree. Ask the teacher for more information.

What is this number at the top of the tree?

Worked Solution
Create a strategy

Multiply the numbers to obtain the values in the factor tree.

Apply the idea
\displaystyle \text{Number}\displaystyle =\displaystyle 2 \times 2 \times 5 \times 5 \times 7 \times 11Multiply the numbers at the end of each branch
\displaystyle \text{ }\displaystyle =\displaystyle 7700Perform the multiplication

Example 2

Write 144 as a product of prime factors in expanded form.

Worked Solution
Create a strategy

Use a factor tree.

Apply the idea
A factor tree for the 144. 144 branches out to 12 and 12. Each 12 branches out to 4 and 3. Each 4 branches out to 2 and 2.

Find two factors of 144 and all composite numbers in the tree: \begin{aligned} 144 &= 12 \times 12 \\ 12 &= 3 \times 4 \\ 4 &= 2 \times 2 \end{aligned}

Using the prime numbers from the tree we get: 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3

Reflect and check

We can also use exponential form to shorten the expression:

\displaystyle 144\displaystyle =\displaystyle 2^3 \times 3^2
Idea summary

A number is a prime if it has exactly two factors: 1, and itself.

A number is composite if it has more than two factors.

A factor tree starts with the number that needs to be investigated and branches out to two factors. Each composite number in the factor tree branches out to two more factors until the last row of the tree are all primes.

What is Mathspace

About Mathspace