# Find Prime Factors

In this worksheet, students will find prime factors of numbers with up to three digits, recording them in both ascending lists and index form.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Number Operations and Integers

Curriculum subtopic:   Structure and Calculation, Whole Number Theory

Difficulty level:

### QUESTION 1 of 10

One of the skills you need to have to be successful in maths is being able to find the prime factors of any number.

This will enable you to solve complex problems in fewer steps and to speed up in your mental maths.

The proper name for this skill is finding or using 'the product of prime factors'.

Let's look at the meaning of each word:

Product: Numbers multiplied together.

Prime: A number that can only be divided by itself and 1 (e.g. 2, 3, 5, 7, 11, 13, 17, 19, etc.)

Factor: A number you can divide a target by that doesn't leave a remainder.

So if we are using the product of prime factor, we are breaking down a number into prime factors and writing them as a list that is then multiplied together.

e.g. We could write 8 as 2 x 2 x 2

This works because 2 is a factor of 8 but it is also a prime number, and we can use this number as a multiple to reach the target number.

How do we do this?

We use a technique called a prime factor tree.

e.g. Find the prime factors of 18.

Once we get to this point, we can write 18 as the product of its prime factors:

18 = 2 x 3 x 3

When we are writing these sums, we always write them in ascending order.

Writing in index form

Another skill we need is to write these answers in index form.

This sounds complicated, but it simply means to write them with powers.

So 18 = 2 x 3 x 3

Can also be written as:

18 = 2 x 32

Be careful if you are asked for index form; you can lose marks if you have the correct answer but miss this step or make an error at this late stage.

In this activity, you will find prime factors of numbers with up to three digits, recording them in both ascending lists and index form.

You will want to grab a pen and scrap paper before you start so you can draw your own prime factor trees.

Which of the prime numbers in the list below can you divide 210 by?

2

3

5

7

11

If you completed a prime factor decomposition such as:

324 = 22 x 34

2

3

5

7

11

Find 180 as a list of prime factors in the form A x B x C x D x E.

Choose the prime number that represents each letter in the table below.

You do not need to consider index form in this question.

Find 180 as a product of prime factors.

Give your answer in index form using the structure of: ab x cd x e.

Choose the number that represents each letter in the table below.

 2 3 5 7 11 a b c d e

Which of the options in the table is a correct product of prime factors for 250?

128 can be written in the form 2a.

Find the value of a.

Match each of the numbers below with their correct product of prime factors.

## Column B

100
22 x 3 x 5
28
22 x 7
60
22 x 52

Which of the options in the list shows the correct product of prime factors for 720?

24 x 32 x 5

24 x 32 x 7

24 x 32 x 11

Write 42 as a product of prime factors.

Do not use index form.

Then complete the sentence below.

• Question 1

Which of the prime numbers in the list below can you divide 210 by?

2
3
5
7
EDDIE SAYS
The only option that doesn't work from the list is 11. For example, if you calculate: 210 ÷ 2 = 105 So 105 and 2 are factors of 210. You can divide 210 by all of the other options in the list. This means they are all prime factors of 210.
• Question 2

If you completed a prime factor decomposition such as:

324 = 22 x 34

EDDIE SAYS
The formal term for powers is indices. So every time you use powers to express a number, you are in fact using index form. It is very common to be asked to write your answer to questions of this type "in index form". Great focus, let's move on!
• Question 3

Find 180 as a list of prime factors in the form A x B x C x D x E.

Choose the prime number that represents each letter in the table below.

You do not need to consider index form in this question.

EDDIE SAYS
Did you draw a prime factor tree here? Remember that you need to continue extending each branch, and dividing each factor, until you reach a prime number. Don't forget that you have to put your prime factors in ascending order. The easiest way to check that you've got them right is just to multiply them all together. If you get back to 180, you've got it right!
• Question 4

Find 180 as a product of prime factors.

Give your answer in index form using the structure of: ab x cd x e.

Choose the number that represents each letter in the table below.

 2 3 5 7 11 a b c d e
EDDIE SAYS
We found in the previous question that: 180 = 2 x 2 x 3 x 3 x 5 So how do we write this in index form? That's right, we need to change the numbers into powers. So we need to identify the numbers which appear more than once which, in this case, is 2 and 3. 2 x 2 = 22 3 x 2 = 32 So our final answer in index form is: 180 = 22 x 32 x 5
• Question 5

Which of the options in the table is a correct product of prime factors for 250?

EDDIE SAYS
If we draw a prime factor tree, we should find that: 250 = 2 x 5 x 5 x 5 As we have more than one 5 in our sum, we can also write this in index form as: 2 x 53 The second and third options are incorrect as they are both missing one x 5, so give an answer of 50 rather than 250. This process is hopefully getting a little clearer for you now, but let's practise some more.
• Question 6

128 can be written in the form 2a.

Find the value of a.

7
EDDIE SAYS
If we draw a prime factor tree and divide each branch until we reach a prime number, we should find that: 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2 How many 2s have been multiplied together here? How do we write this as a power? We have 7 2s present, which we can write as: 27
• Question 7

Match each of the numbers below with their correct product of prime factors.

## Column B

100
22 x 52
28
22 x 7
60
22 x 3 x 5
EDDIE SAYS
The best way to approach this matching is by a process of elimination. All the numbers are in the 4 times table, so we can substitute the number 4 for 22 to make life easier. 28 can be written as 4 x 7 100 can be written as 4 x 25 60 can be written as 4 x 15 Hopefully, this makes each number easier to match, as you just need to consider the second half of each product.
• Question 8

Which of the options in the list shows the correct product of prime factors for 720?

24 x 32 x 5
EDDIE SAYS
Did you do the decomposition for each sum? This means expanding each one out and calculating the answer e.g. 24 x 32 x 7 = 16 x 9 x 7 = 1008 As this answer is not 720, we know that this option is not correct. Nothing wrong with that if you used the method above, but there is a quicker way here. Can you divide 720 by 7 or 11? If the answer is no (which it is!), then you can disregard these sums with no further calculation required. Doesn't that make life easier?!
• Question 9

Write 42 as a product of prime factors.

Do not use index form.

2x3x7
2 x 3 x 7
EDDIE SAYS
If you draw a prime factor tree, you will reach 2, 3 and 7 as your prime factors. Don't forget that you have to write the factors in ascending order to get the marks.
• Question 10

Then complete the sentence below.

EDDIE SAYS
You can only put prime factors into index form if they appear more than once as a product. 42 = 2 x 3 x 7 So there are no repeating prime factors, which means you cannot put it in index form. Great work on completing this activity! Hopefully you are feeling much more confident to recognise and find prime factors now.
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