Extension: Prime factorizations

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.

Here is how we might start with the number 12.

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

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

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

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.

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.

Here is a factor tree for 360:

We can therefore write:

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:

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 11	Multiply the numbers at the end of each branch
\displaystyle \text{ }	\displaystyle =	\displaystyle 7700	Perform the multiplication

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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

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