Factors, multiples and primes Revision Notes for Edexcel GCSE Maths

Factors, multiples and primes

What are factors?

Factors are whole numbers that divide into another number exactly, leaving no remainder. Every number has at least two factors: 1 and the number itself.

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For example, the factors of 12 are: 1, 2, 3, 4, 6, and 12. Notice how each of these numbers divides into 12 with no remainder.

Factor pairs

Factors naturally come in factor pairs. Each pair represents a multiplication that gives the original number as the answer.

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Worked Example: Factor Pairs of 12

The factor pairs of 12 are:

$1 \times 12 = 12$
$2 \times 6 = 12$
$3 \times 4 = 12$

Each pair multiplies together to give 12.

Common factors

A common factor is a number that is a factor of two or more numbers. For instance, 2 is a common factor of both 6 and 12 because it divides into both numbers exactly.

What are multiples?

Multiples of a number are all the numbers that appear in its times table. These are the results when you multiply the number by 1, 2, 3, 4, and so on.

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The multiples of 7 are: 7, 14, 21, 28, 35, 42, ... and they continue infinitely.

Common multiples

A common multiple is a number that appears in the times tables of two or more numbers. For example, 12 is a common multiple of both 6 and 4.

Prime numbers

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A prime number has exactly two factors: 1 and itself. This means it can only be divided evenly by these two numbers.

Critical point: The number 1 is not a prime number because it only has one factor (itself), not two.

The first ten prime numbers

You should memorise these: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Factor trees

Factor trees help you find the prime factors of any number. This method breaks down a number into its prime components.

Steps to create a factor tree

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Worked Example: Creating a Factor Tree

Choose any factor pair of the number
Circle the prime factors as you find them
Continue breaking down non-prime factors until every branch ends with a prime number
Write all the circled prime numbers as a multiplication

Example: Factor tree for 90

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Worked Example: Factor Tree for 90

$90 = 5 \times 2 \times 3 \times 3 = 5 \times 2 \times 3^2$

When 3 appears twice in the factor tree, write it as $3^2$ in index form.

Worked examples

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Worked Example 1: Identifying from a List

From the numbers: 16, 8, 3, 17, 6, 20, 12

Prime number: 17 (only divisible by 1 and 17)
Multiple of 5: 20 (appears in the 5 times table)
Two factors of 24 with sum 15: 12 and 3 (since $12 \times 2 = 24$ , $3 \times 8 = 24$ , and $12 + 3 = 15$ )

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Worked Example 2: Prime Factorisation

To write 280 as a product of prime factors:

Use a factor tree to break down 280
Circle each prime factor as you find it
Express repeated prime factors using index form
Check your answer by multiplying the prime factors back together

Exam tips

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Key Exam Strategies:

Recognise prime numbers below 50 quickly - this saves time in exams
Check your factor trees by multiplying your final answer
Look carefully at what the question asks for - prime numbers, factors, or multiples
Show your working clearly, especially for factor trees

Remember!

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Key Points to Remember:

Factors divide into a number exactly - they include 1 and the number itself
Multiples are numbers in a times table - they go on forever
Prime numbers have exactly two factors: 1 and themselves
Factor trees help find prime factors by repeatedly dividing by prime numbers
Index form (like $3^2$ ) shows when prime factors repeat in factor trees

Factors, multiples and primes (Edexcel GCSE Maths): Revision Notes