Factors, Multiples, Primes, HCF and LCM

This lesson covers key topics from the AQA GCSE Mathematics Number specification: identifying factors, multiples, and prime numbers, and using prime factorisation to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM). These concepts are tested regularly on both papers and form the building blocks for work with fractions, algebra and more.

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Factors

A factor of a number is any whole number that divides into it exactly, leaving no remainder.

Finding Factors

To find all the factors of a number, work through pairs systematically starting from 1.

Worked Example: Find all the factors of 36.

Pair	Calculation
1 x 36	1 and 36 are factors
2 x 18	2 and 18 are factors
3 x 12	3 and 12 are factors
4 x 9	4 and 9 are factors
6 x 6	6 is a factor (pair repeats)

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Exam Tip: Always start with 1 and the number itself, then work inward. When your factor pairs start repeating (or meet in the middle), you have found them all.

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Multiples

A multiple of a number is the result of multiplying that number by a positive integer. Multiples are like entries in a times table.

First ten multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

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Prime Numbers

A prime number has exactly two factors: 1 and itself.

Number	Factors	Prime?
1	1	No — only one factor
2	1, 2	Yes — the only even prime
3	1, 3	Yes
4	1, 2, 4	No — three factors
5	1, 5	Yes
9	1, 3, 9	No — three factors
11	1, 11	Yes
15	1, 3, 5, 15	No

Key primes to know: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Exam Tip: Remember that 1 is NOT a prime number and that 2 is the only even prime number. These are very common exam traps.

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Prime Factorisation

Prime factorisation means writing a number as a product of its prime factors.

Method 1: Factor Trees

A factor tree breaks a number down by dividing by prime factors until every branch ends in a prime.

graph TD
    A[60] --> B[2]
    A --> C[30]
    C --> D[2]
    C --> E[15]
    E --> F[3]
    E --> G[5]

Reading the prime factors from the tree: 60 = 2 x 2 x 3 x 5 = 2 squared x 3 x 5

Method 2: Repeated Division

Divide by the smallest prime factor repeatedly:

Step	Division	Result
1	60 / 2	30
2	30 / 2	15
3	15 / 3	5
4	5 / 5	1

60 = 2 x 2 x 3 x 5 = 2 squared x 3 x 5

Another Worked Example

Write 180 as a product of its prime factors.

graph TD
    A[180] --> B[2]
    A --> C[90]
    C --> D[2]
    C --> E[45]
    E --> F[3]
    E --> G[15]
    G --> H[3]
    G --> I[5]

180 = 2 x 2 x 3 x 3 x 5 = 2 squared x 3 squared x 5

Exam Tip: Always write your final answer using index notation (e.g. 2 squared x 3 squared x 5) unless the question says otherwise. This is the expected format for full marks on AQA papers.

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Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that is a factor of all of them.

Method: Using Prime Factorisation and Venn Diagrams

Find the HCF of 60 and 84.

Step 1: Prime factorise both numbers.

• 60 = 2 x 2 x 3 x 5
• 84 = 2 x 2 x 3 x 7

Step 2: Draw a Venn diagram. Place shared prime factors in the overlap.

graph TD
    subgraph Venn Diagram
        direction LR
        A["60 only: 5"]
        B["Overlap: 2, 2, 3"]
        C["84 only: 7"]
    end

Step 3: Multiply the numbers in the overlap.

HCF = 2 x 2 x 3 = 12

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Lowest Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Method: Using the Venn Diagram

Using the same Venn diagram from the HCF example above:

Step: Multiply ALL the numbers in the Venn diagram (both circles, including the overlap).

LCM = 5 x 2 x 2 x 3 x 7 = 420

Alternative Method: Listing Multiples

For smaller numbers, you can list multiples:

Find the LCM of 6 and 8.

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48 ...
• Multiples of 8: 8, 16, 24, 32, 40, 48 ...

LCM = 24

Exam Tip: The Venn diagram method is the most efficient for larger numbers and is the method AQA expects. Practise it until it becomes second nature.

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Combined Worked Example

Find the HCF and LCM of 120 and 150.

Step 1: Prime Factorise

• 120 = 2 x 2 x 2 x 3 x 5 = 2 cubed x 3 x 5
• 150 = 2 x 3 x 5 x 5 = 2 x 3 x 5 squared

Step 2: Venn Diagram

120 only	Overlap	150 only
2, 2	2, 3, 5	5

Step 3: Calculate

• HCF = 2 x 3 x 5 = 30
• LCM = 2 x 2 x 2 x 3 x 5 x 5 = 600