Factors, multiples and primes | AQA GCSE Maths Notes

Visual model

Prime factor trees break numbers into building blocks

60

6

10

keep splitting until every end is prime

Gold-standard guide

20 mins

What you will learn

Use prime factors, HCF and LCM to solve number problems.

Use a clear step-by-step method for factors, multiples and primes.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier number skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Prime factor trees give you both HCF and LCM from one $pie$ ce of working

Step 1

Write 60 as a product of prime factors

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Step 2

Write 84 as a product of prime factors

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7

Step 3

Find the HCF — use only shared factors with their lower powers

Both share 2² and 3¹

Watch out

Students confuse HCF and LCM

HCF

Use the shared prime factors with the lowest powers.

LCM

Use every prime factor needed, with the highest powers.

Worked example

Find the HCF and LCM of 60 and 84 using prime factor decomposition

Write 60 as a product of prime factors: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Write 84 as a product of prime factors: 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7.

Find the HCF — use only shared factors with their lower powers: Both share 2² and 3¹. HCF = 2² × 3 = 12.We use the LOWER power of each shared factor.

Find the LCM — take all factors with their higher powers: Factors present: 2² (appears in both, take highest power = 2²), 3¹, 5¹, 7¹.LCM = 4 × 3 × 5 × 7 = 420.

Final answer

HCF = 12, LCM = 420

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Find the HCF and LCM of 60 and 84 using prime factor decomposition

4 marks4 minsfactors-multiples-and-primes-worked

Show solution

Worked solution

1.Write 60 as a product of prime factors: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
2.Write 84 as a product of prime factors: 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
3.Find the HCF — use only shared factors with their lower powers: Both share 2² and 3¹. HCF = 2² × 3 = 12.We use the LOWER power of each shared factor.
4.Find the LCM — take all factors with their higher powers: Factors present: 2² (appears in both, take highest power = 2²), 3¹, 5¹, 7¹.LCM = 4 × 3 × 5 × 7 = 420.

Final answer

HCF = 12, LCM = 420

Mark points

M1: write 60 as a product of prime factors
M1: write 84 as a product of prime factors
M1: find the hcf — use only shared factors with their lower powers
M1: find the lcm — take all factors with their higher powers
A1: HCF = 12, LCM = 420

Watch out

Students confuse HCF and LCM.HCF is the biggest number that divides BOTH numbers (it will be smaller than or equal to both).LCM is the smallest number that BOTH numbers divide into (it will be larger than or equal to both).If your HCF is bigger than one of the numbers, or your LCM is smaller than one of them, you have swapped the methods.

Diagnosticrecall

List all the factors of 36

1 mark2 minsfactors-multiples-and-primes-q1

Show solution

Worked solution

1.Spot the skill: Prime factor trees give you both HCF and LCM from one $pie$ ce of working.
2.Use the write 60 as a product of prime factors stage first, then write 84 as a product of prime factors.
3.Keep the final answer visible: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Final answer

1, 2, 3, 4, 6, 9, 12, 18, 36

Mark points

M1: use the correct prime factor trees give you both hcf and lcm from one $pie$ ce of working.hcf = multiply the shared prime factors. lcm = multiply all prime factors (shared ones counted once).think: hcf = intersection, lcm = union.
A1: 1, 2, 3, 4, 6, 9, 12, 18, 36

Watch out

Easyprocedure

Write 180 as a product of prime factors

2 marks3 minsfactors-multiples-and-primes-q2

Show solution

Worked solution

1.Spot the skill: Prime factor trees give you both HCF and LCM from one $pie$ ce of working.
2.Use the write 84 as a product of prime factors stage first, then find the hcf — use only shared factors with their lower powers.
3.Keep the final answer visible: 2² × 3² × 5.

Final answer

2² × 3² × 5

Mark points

M1: use the correct prime factor trees give you both hcf and lcm from one $pie$ ce of working.hcf = multiply the shared prime factors. lcm = multiply all prime factors (shared ones counted once).think: hcf = intersection, lcm = union.
A1: 2² × 3² × 5

Watch out

Mediumreasoning

Find the HCF of 48 and 72

3 marks4 minsfactors-multiples-and-primes-q3

Show solution

Worked solution

1.Spot the skill: Prime factor trees give you both HCF and LCM from one $pie$ ce of working.
2.Use the find the hcf — use only shared factors with their lower powers stage first, then find the lcm — take all factors with their higher powers.
3.Keep the final answer visible: 24.

Final answer

Mark points

M1: use the correct prime factor trees give you both hcf and lcm from one $pie$ ce of working.hcf = multiply the shared prime factors. lcm = multiply all prime factors (shared ones counted once).think: hcf = intersection, lcm = union.
A1: 24

Watch out

Hardproblem solving

Find the LCM of 8, 12 and 15

3 marks5 minsfactors-multiples-and-primes-q4

Show solution

Worked solution

1.Spot the skill: Prime factor trees give you both HCF and LCM from one $pie$ ce of working.
2.Use the find the lcm — take all factors with their higher powers stage first, then write 60 as a product of prime factors.
3.Keep the final answer visible: 120.

Final answer

120

Mark points

M1: use the correct prime factor trees give you both hcf and lcm from one $pie$ ce of working.hcf = multiply the shared prime factors. lcm = multiply all prime factors (shared ones counted once).think: hcf = intersection, lcm = union.
A1: 120

Watch out

Exam-stylemulti-step

Two lighthouses flash every 24 seconds and every 36 seconds respectively. They flash together at midnight. How many seconds until they next flash together?

4 marks6 minsfactors-multiples-and-primes-q5

Show solution

Worked solution

1.Spot the skill: Prime factor trees give you both HCF and LCM from one $pie$ ce of working.
2.Use the write 60 as a product of prime factors stage first, then write 84 as a product of prime factors.
3.Keep the final answer visible: 72 seconds.

Final answer

72 seconds

Mark points

M1: use the correct prime factor trees give you both hcf and lcm from one $pie$ ce of working.hcf = multiply the shared prime factors. lcm = multiply all prime factors (shared ones counted once).think: hcf = intersection, lcm = union.
A1: 72 seconds

Watch out

Grade 9 stretchproblem solving

Find the smallest positive integer n such that 72n is a square number.

4 marks7 minsprime-g9

Show solution

Worked solution

1.Write 72 as a product of prime factors.
2.Make every power even using the smallest possible multiplier.

Final answer

n = 2

Mark points

M1: 72 = 2³ × 3²
A1: multiply by 2

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Hard exam-stylemulti-step problem

Find the smallest positive integer k such that 540k is a cube number.

3 marks6 minsprime-paper

Show solution

Worked solution

1.Write 540 as a product of prime factors.
2.For a cube, each power must be a multiple of 3.
3.Choose the smallest missing factors.

Final answer

k = 50

Mark points

M1: use 540 = 2² × 3³ × 5
M1: identify one more factor 2 and two more factors 5
A1: obtain k = 50

Watch out

Read the full question before calculating. Keep each stage of your working visible.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Factors, multiples and primes - 2 marksList all the factors of 36Mark answer

Answer

1, 2, 3, 4, 6, 9, 12, 18, 36

2Calculations and order of operations - 2 marksWork out (6 + 2) × 3 − 5Mark answer

Answer

3Integers, decimals and place value - 2 marksOrder these from smallest to largest: 0.3, 0.03, 0.303, 0.033Mark answer

Answer

0.03, 0.033, 0.3, 0.303

4Fractions - 3 marksA recipe needs ¾ cup of sugar. How much sugar is needed for 2½ batches?Mark answer

Answer

1⅞ cups

Mastery check

I can explain the method for factors, multiples and primes.
I can show clear working without skipping key steps.
I can avoid this mistake: Students confuse HCF and LCM.HCF is the biggest number that divides BOTH numbers (it will be smaller than or equal to both).LCM is the smallest number that BOTH numbers divide into (it will be larger than or equal to both).If your HCF is bigger than one of the numbers, or your LCM is smaller than one of them, you have swapped the methods.

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