1.11 Prime factorisations

One of the best ways to find a prime factorisation 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 make12, 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 = 12 \times 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 factorisation 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

Watch question walkthrough

Example 2

Write 148 as a product of prime factors in expanded form.

Worked Solution

Create a strategy

Start with dividing the number with the smallest possible prime number and repeating the same process with its quotient and so on until the quotient becomes a prime number.

Apply the idea

\displaystyle 148\div 2	\displaystyle =	\displaystyle 74	Divide by 2
\displaystyle 74 \div 2	\displaystyle =	\displaystyle 37	Divide the quotient by 2

Since 37 is a prime number, we stop here. So we now have the expanded form: 148= 2 \times 2 \times 37

Watch question walkthrough

Factor trees are useful because every number we write as we make it is a factor of the original number. We do not always see every factor appear, though - for example, 9 is a factor of 360, but it does not appear in the tree above.

To find every factor of a number we need to combine the prime factors in every possible way. First, we find the prime factorisation like we did before, such as:

12 = 2 \times 2 \times 3

We then combine these factors in every possible way. Every factor of 12 can have no 2s, one 2, or two 2s in its prime factorisation. Similarly, every factor of 12 can have no 3s, or one 3. Here we draw this out in a table:

We then perform each of the multiplications to find all the factors:

The factors of 12 are 1,\,2,\,3,\,4,\,6, and 12.

Here is how we can do it for 360 - every factor either has 5 as a factor or it doesn't, it has between zero and two 3s, and between zero and three 2s.

This table shows all the possible ways to multiply the prime factors together. We evaluate the multiplications to find all the factors:

The factors of 360 are 1, 2,\,3,\,4,\,5,\,6,\,8,\,9,\,10,\,12,\,15,\,18,\,20,\,24,\,30,\,36,\,40,\,45,\,60,\,72,\,90,\,120,\,180 and 360.

Examples

Example 3

Find the factors of 20.

a

Write 20 as a product of prime factors in expanded form.

Worked Solution

Create a strategy

Draw a factor tree to obtain the factors.

Apply the idea

Here is the beginning of a factor tree for \\ 20 = 4 \times 5

Continue the factor tree by finding factors of 4, and leaving the 5 which is a prime number.

4 = 2 \times 2

So:20=2\times2\times5

Watch question walkthrough

b

Using your answer from part (a), list all the factors of 20.

Worked Solution

Create a strategy

We can multiply all combinations of the prime factors to find all the factors.

Apply the idea

	\text{No }2	\text{One }2	\text{Two }2\text{s}
\text{No }5	1\times 1=1	1\times 2=2	2\times 2=4
\text{One }5	1\times 5=5	2\times 5=10	2\times 2\times 5=20

So the factors of 20 are 1,\,2,\,4,\,5,\,10,\,20.

Watch question walkthrough