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Prime factors Factor treesFactors and multiples

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Here we will learn about the **highest common factor **including how to calculate the highest common factor, use the highest common factor, and recognise when to calculate the highest common factor to answer complex worded problems.

There are also highest common factor worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

The **highest common factor** (HCF) or greatest common factor is the largest integer that two or more numbers can be divided by.

**Highest **meaning largest number or greatest.**Common **meaning shared between two or more numbers.**Factor **meaning an integer that a whole number can be divided by (a divisor).

There are several interchangeable terms you should be aware of:

HCF represents the highest common factor, GCF represents the greatest common factor, and GCD represents the greatest common divisor. These are all the same!

E.g.

Find the HCF of

Let’s start by writing the factors of 4 and 6,

We can see that the highest number that occurs in each list is

hence for the numbers **highest common factor** is ** 2**.

NOTE:

Calculating the **highest common factor** becomes more complicated for larger numbers as listing all the factors of each number is very time consuming.

To make it easier we can utilise the **prime factors** of the numbers.

The fundamental theorem of arithmetic states that each number greater than

This means that **every number has a unique set of prime factors**.

We can use **prime factors** and a **Venn diagram** to calculate the **highest common factor.**

To do this we need to write out the prime factorisation of each number fully (without any powers) and then put the numbers into the Venn diagram by looking for pairs.

Here, we have the prime factors of

- In the left circle are the prime factors of
12 (2 ,2 , and3 ). - In the right circle, we have the prime factors of
30 (2 ,3 , and5 ). - The dark orange section on the left represents the prime factors of
12 that are not prime factors of30 (which is just the value2 ). - The light orange section on the right represents the prime factors of
30 that are not prime factors of12 (which is just the value5 ). - The intersection represents the prime factors of
12 and30 .

Here, the **prime factors** of **2** and **3** occur for both 12 and 30 so their** product is the highest common factor**.

The **highest common factor** for ** 12 **and

The reason why the intersection contains the highest common factor is because there are no other common prime factors in this set of numbers, otherwise they would be in the intersection.

You must remember to write every factor out separately into the Venn diagram.

If you have more than

In order to calculate the highest common factor for two or more numbers:

**State the product of prime factors for each number, not in index form****(you can work these out using a prime factor tree)****Write all the prime factors into the Venn diagram for each number****Multiply the prime factors in the intersection to find the HCF**

Get your free highest common factor worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free highest common factor worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONCalculate the highest common factor of

**State the product of prime factors for each number, not in index form**

2**Write all the prime factors into the Venn diagram for each number**

3**Multiply the prime factors in the intersection to find the HCF**

HCF = **6**

Calculate the highest common factor of

**State the product of prime factors for each number, not in index form**

**Write all the prime factors into the Venn diagram for each number**

**Multiply the prime factors in the intersection to find the HCF**

HCF = **14**

Given that ^{2} × 5

**State the product of prime factors for each number, not in index form**

To state the product of prime factors of

**Write all the prime factors into the Venn diagram for each number**

**Multiply the prime factors in the intersection to find the HCF**

HCF = **18**

Calculate the highest common factor of

**State the product of prime factors for each number, not in index form**

**Write all the prime factors into the Venn diagram for each number**

**Multiply the prime factors in the intersection to find the HCF**

HCF =

Calculate the highest common factor of

**State the product of prime factors for each number, not in index form**

**Write all the prime factors into the Venn diagram for each number**

**Multiply the prime factors in the intersection to find the HCF**

Here there are no prime factors in the intersection, so the highest common factor is equal to **1**

Sue sells apples in baskets. On Thursday, she will receive a delivery of

**State the product of prime factors for each number, not in index form**

**Write all the prime factors into the Venn diagram for each number**

**Multiply the prime factors in the intersection to find the HCF**

HCF **6**

So there will be **6****apples **in each basket.

**Calculating the Lowest Common Multiple (lcm) instead**

A very common misconception is mixing up the highest common factor with the lowest common multiple.

Remember:

Factors are composite numbers that are split into smaller factors

Multiples are composite numbers that are multiplied to make larger multiples

**Incorrect evaluation of powers**

It is possible to write prime factors into a Venn diagram with their associated exponent or power. This becomes an issue when the powers are not correctly interpreted.

Take for example the numbers

^{3}

^{4}

This means that we have common factors of

The correct answer is

The remaining factor of

The Venn diagram would therefore look like this:

It is recommended that you write out each product of prime factors without using index form as this will make writing the numbers into the Venn diagram easier.

1. Calculate the highest common factor of 22 and 60 .

1

660

11

2

22= 2 \times 11 \\
60= 2 \times 2 \times 3 \times 5
HCF = 2

2. Calculate the highest common factor of 21 and 63 .

3

7

21

63

21=3 \times 7\\
63 = 3 \times 3 \times 7
\text{HCF }=7 \times 3=21

3. Calculate the highest common factor of 90 and 135 .

45

15

1

270

90= 2 \times 3 \times 3 \times 5 \\
135=3 \times 3 \times 3 \times 5
\text{HCF }=3\times 3 \times 5=45

4. Calculate the highest common factor of 12 , 30 and 48 .

12

2

3

6

12=2 \times 2 \times 3\\
30=2 \times 3 \times 5\\
48=2 \times 2 \times 2 \times 2 \times 3
\text{HCF }= 2 \times 3=6

5. Calculate the highest common factor of 21 and 58 .

0

1

3

1218

21=3 \times 7\\
58= 2 \times 29
HCF=1

6. Josh is grouping 3 different coloured writing pens. He has 18 blue, 16 black, and 28 red. Each group must contain the same amount of each colour pen. How many groups can Josh make, so that every pen has been grouped?

2

1

4

1008

We need to calculate the HCF of 18, 16 and 28. \\18=2 \times 3 \times 3\\ 16=2 \times 2 \times 2 \times 2\\ 28=2 \times 2 \times 7 \\ HCF=2

`Therefore Josh can make 2 groups of pens with the same number of each colour in each group.`

1. The highest common factor of a and b is 5 . The lowest common multiple of a and b is 30 . State the values of a and b .

**(4 marks)**

Show answer

**(1)**

30 \div 5=6

**(1)**

` 6= 2 \times 3 `

**(1)**

a=10 and b = 15

**(1)**

(accept the alternative solution: a=15 and b=10 )

2.

(a) Calculate the highest common factor of the two numbers a=16 g^{2} h^{2} \text { and } b=28 g^{3} h

(b) Let g=3 and h=5

(i) Calculate the values of a and b

(ii) Hence or otherwise, calculate the value of the highest common factor when g=3 and h=5

**(8 marks)**

Show answer

\text { a) }16 g^{2} h^{2}=2 \times 2 \times 2 \times 2 \times g \times g \times h \times h \text { or } 28g^{3}h=2 \times 2 \times 7 \times g \times g \times g \times h

**(1)**

**(1)**

**(1)**

**(1)**

**(1)**

**(1)**

**(1)**

**(1)**

3. A school is conducting a research project. Year 11 classes will be split into small groups with the same proportion of students from each class.

- Class 11 R has 32 pupils
- Class 11 B has 28 pupils
- Class 11 G has 36 pupils.

What is the maximum number of groups possible?

**(3 marks)**

Show answer

32=2 \times 2 \times 2 \times 2 \times 2 \text { or } 32=2^{5}

**(1)**

**(1)**

**(1)**

You have now learned how to:

- Use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property

- Lowest common multiple

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