# Combinations

## Topic Overview

During your GCSE Maths exam, you will be required to answer a series of questions regarding the combinations of different values. In order to do this, you will need to understand the basic principles of factors in order to solve more complicated problems.

The **factors** of a number are any numbers which divide into it exactly. These factors include 1 and the number itself.

For example, the factors of 6 are: 1, 2, 3 and 6.

When identifying the factors of larger numbers it is often easier to 'pair' the factors by writing them as multiplications.

For example, the factors of 24 are:

1 x 24 = 2 x 12 = 3 x 8 = 4 x 6

So the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

## Key Concepts

In the new linear GCSE Maths paper, you will be required to work out several problems involving combinations. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

- Use the concepts and vocabulary of factor (divisor), multiple and common factor
- Use the concepts of highest common factor, least common multiple, prime number and prime factor decomposition

Listed below are a series of summaries and worked examples to help you solidify your knowledge about these different types of combinations.

**1 - Factors**

During your exam, you will be asked to work out the factors of certain values.

*Example*

(a) - What are the factors of 50?

*Solution*

(a) - It is often easier to pair up the factors of a certain value:

50= 1x50 = 2x25 = 5x10

Therefore the factors of 50 are: 1, 2, 5, 10, 25 and 50.

**2 - Highest common factor (HCF)**

The **Highest Common Factor**, or HCF as it is commonly known, is the largest positive integer which divides two or more integers without any remainder. For example:

When comparing the factors of 24 and 40:

The factors of 24 are : **1**, **2**, 3, **4**, 6, **8**, 12 and 24

The factors of 40 are: **1**, **2**, **4**, 5, **8**, 10, 20 and 40.

When comparing both lists, both 24 and 40 share the factors : 1, 2, 4 and 8. These factors are referred to as the **common factors** of 24 and 40.

The number 8 is the highest of these common factors, making it the highest common factor or HCF of 24 and 40.

*Example*

(a) - What is the HCF of 20 and 40?

*Solution*

(a) - The factors of 20 are: 1, 2, 4, 5, 10 and 20

The factors of 40 are: 1, 2, 4, 5, 8, 10, 20 and 40

The common factors of 20 and 40 are 1, 2, 4, 5 ,10 and 20

Therefore the HCF of 20 and 40 is 20.

**3 - Least Common Multiples (LCM)**

When you list the multiples of two (or more) numbers, you will often find the same value in both lists. This value is known to be a common multiple of those numbers.

For example, if you were asked to compare the multiples of 4 and 5, the common multiples which are found in both lists are:

Multiples of 4: 4, 8, 12, 16, **20**, 24, 28, 32, 36, **40** ,44 etc...

Multiples of 5: 5, 10, 15, **20** ,25, 30 ,35, **40**, 45 etc...

Therefore the common multiples of 4 and 5 are: 20 and 40 etc...

Subsequently, the **Least Common Multiple**, or **LCM** as it is commonly known, is the smallest value out of these common multiples, which in this example is 20.

*Example*

(a) - What is the LCM of 8 and 40?

*Solution*

(a) - The multiples of 8 are 8, 16, 24, 32, **40**, 48 etc...

The multiples of 20 are 20, **40**, 60, 80, 100, 120 etc...

Therefore the LCM of 8 and 20 is: 40.

**4 - Prime Factors**

In order to understand prime factors, you must first understand prime numbers.

A **prime number** can only be divided evenly by 1 or itself and it must be a whole number greater than 1.

For example: 2 can only be divided evenly by 1 or 2, so it is a prime number.

Every number can be written as a product of prime numbers.

Therefore a **prime factor** is a factor that is a prime number. It is one of the prime numbers that, when multiplied, give the original number.

For example, the prime factors of 15 are 3 and 5. This is because 3×5=15, and 3 and 5 are both prime numbers.

During your exam, you may be asked to write a number as a product of its prime factors. In these cases, it is important to remember that 'product' means 'times' or 'multiply'. Therefore, if you were asked to write 40 as a product of its prime factors, it would be:

40 = 2 x 2 x 2 x 5

There are several methods through which you can write a number as a product of its prime factors. The following examples will introduce you to these different methods:

*Example*

(a) - Write 24 as a product of its prime factors

*Solution*

(a) - Method One

- Begin with the smallest prime number which divides into 24

24= 2x 12 (Note: 2 is the smallest prime number for all even numbers)

- Now write down the smallest prime number which divides into 12.

24 = 2 x 2 x 6 (Once again 2 is the smallest prime number for all even numbers)

- Continue this method until you have written down all of the prime numbers for the value in question

6= 2 x 3

Therefore 24= 2 x 2 x 2 x 3

(a) Method Two

- Use a factor tree

A '**factor tree**' is a diagram which is used to break down a number by dividing it by its factors until all the numbers left are prime. A factor tree follows the same process as method one, but displays the process in a more visible format.

Subsequently, you can see that 24 = 2 x 2 x 2 x 3

In cases such as these where you have multiple prime factors which are the same, you can write these factors in index form. Therefore, in this example:

24 = 2 x 2 x 2 x 3, which becomes :

24 = 2³ x 3

Once you have understood the principles of prime factors, you can then use them to find the HCF and LCM of other values.

*Example*

(b) - Find the HCF of 24 and 36

*Solution*

(b) - You can use prime factors to calculate the HCF by following this method:

- List of prime factors of 24 and 36 and locate the common factors:

24 = **2** x **2** x 2 x **3**

36 = **2** x **2** x **3** x 3

- Write down these common factors, and multiply them to find the HCF of 24 and 36:

HCF of 24 and 36 = 2 x 2 x 3 = 12

*Example*

(c) - Find the LCM of 24 and 36

*Solution*

(c) - You can use prime factors to calculate the LCM by following this method:

- List the prime factors of 24 and 36 and locate the common multiples:

24= 2x2x2x3

36= 2x2x3x3

- To calculate the LCM, find out which value has the most of each prime factor and group them together:

24 = (2 x 2 x 2) x (3)

36 = (2 x 2) x (3 x 3)

Subsequently, you can see that 24 has the most 2s (it has three) and 36 has the most 3s (it has two).

Therefore the LCM of 24 and 36 = (2 x 2 x 2) x (3 x 3) = 72.

## Exam Tips

- When faced with a factor related question, first list all of the common factors and multiples of the two values you have been given. This will make it easier for you to compare and calculate the HCF and LCM of these values
- Remember that a prime number is a value which can only be divided by itself and 1
- Remember that 2 is the smallest prime number for all even numbers
- Display all of your working out in order to score maximum method marks

## Topic Summary

Combinations is a topic which depends heavily on displaying your working out. If you write down all aspects of your method, no matter how simple it may appear, you will be able to easily work out more complex questions in your exam.

Remember to group together similar prime factors and display these factors in numerical order to make your working more clear and easy to interpret. By writing clearly, you improve your own understanding of the question and also demonstrate your mathematical knowledge to the examiner who will be marking your paper.

## Related Topics

- Arithmetic with Fractions
- Indices
- Problem Solving with Decimals and Percentages
- Accuracy of Measurement Problems
- Arithmetic with Positive and Negative Integers
- Standard Form
- Rational and Irrational Numbers