N4: Prime Numbers, Factors & Multiples
Prime numbers, factors, multiples, HCF, LCM, prime factorisation, product notation
📋 Key Concepts
Prime Number: A number greater than 1 that has exactly two factors: 1 and itself.
First 20 Prime Numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
Remember: 1 is NOT a prime number. 2 is the only even prime number.
Definitions
| Term | Definition | Example |
|---|---|---|
| Factor | A number that divides exactly into another number | Factors of 12: 1, 2, 3, 4, 6, 12 |
| Multiple | The result of multiplying a number by an integer | Multiples of 5: 5, 10, 15, 20... |
| HCF | Highest Common Factor | HCF of 12 and 18 = 6 |
| LCM | Lowest Common Multiple | LCM of 4 and 6 = 12 |
📝 Finding Factors
Method: Find factor pairs by dividing. Work from 1 upwards until you meet in the middle.
Example 1
Find all factors of 24
Solution:
1 × 24 = 24 → factors: 1, 24
2 × 12 = 24 → factors: 2, 12
3 × 8 = 24 → factors: 3, 8
4 × 6 = 24 → factors: 4, 6
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
📝 Prime Factorisation
Method: Use a factor tree to break down a number into its prime factors. Write using index notation.
Example 2
Express 60 as a product of its prime factors
Solution:
60
/ \
6 10
/ \ / \
2 3 2 5
Prime factors: 2 × 2 × 3 × 5 = 2² × 3 × 5
Example 3
Express 84 as a product of its prime factors
Solution: 84 = 2² × 3 × 7
Check: 4 × 3 × 7 = 84 ✓
📝 Finding HCF
Method 1 - Lists: List all factors of each number and find the largest common one.
Method 2 - Prime Factors: Multiply together the common prime factors with their lowest powers.
Method 2 - Prime Factors: Multiply together the common prime factors with their lowest powers.
Example 4
Find the HCF of 24 and 36
Solution (using prime factors):
24 = 2³ × 3
36 = 2² × 3²
Common factors: 2² × 3 = 12
HCF = 12
📝 Finding LCM
Method 1 - Lists: List multiples of each number until you find the first common one.
Method 2 - Prime Factors: Multiply together all prime factors with their highest powers.
Method 2 - Prime Factors: Multiply together all prime factors with their highest powers.
Example 5
Find the LCM of 24 and 36
Solution (using prime factors):
24 = 2³ × 3
36 = 2² × 3²
LCM = 2³ × 3² = 8 × 9 = 72
📝 HCF and LCM Together
Useful relationship: For any two numbers a and b: HCF × LCM = a × b
Example 6
Two numbers have HCF = 4 and LCM = 60. If one number is 12, find the other.
Solution:
HCF × LCM = a × b
4 × 60 = 12 × b
240 = 12 × b
b = 240 ÷ 12 = 20
🧮 Worked Examples — Exam Style
Prime factorisation is the Swiss-army knife: HCF = lowest powers of COMMON primes; LCM = highest powers of ALL primes.
Express 180 as a product of prime factors.
180 = 2 × 90 = 2 × 2 × 45 = 2² × 3² × 5
M1 for the factor tree/divisions, A1 for 2² × 3² × 5. Present the final line in index form — that is what 'product of primes' means.
Two buses both leave at 09:00. One every 12 minutes, one every 15. When do they next leave together?
LCM of 12 and 15: 12 = 2² × 3, 15 = 3 × 5 → LCM = 2² × 3 × 5 = 60 minutes
Next together: 10:00
M1 for the LCM calculation, A1 for 10:00. LCM questions are always scheduling/timing contexts.
Exam wording to expect: "express as a product of prime factors", "find the HCF/LCM", or a context (buses, tiles, lights flashing together).
❓ Practice Questions
Q1: List all factors of 18
Q2: Express 72 as a product of its prime factors
Q3: Find the HCF of 28 and 42
Q4: Find the LCM of 8 and 12
Q5: Show that 87 is not a prime number
Q6: Find the HCF and LCM of 36 and 48
Q7: Two lights flash every 8 s and 12 s. They flash together now. When do they next flash together?
✅ Answers
- 1, 2, 3, 6, 9, 18
- 72 = 2³ × 3²
- 28 = 2² × 7, 42 = 2 × 3 × 7, HCF = 14
- LCM = 24 (or: 8 = 2³, 12 = 2² × 3, LCM = 2³ × 3 = 24)
- 87 = 3 × 29, so it has more than two factors
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 1, 2, 3, 6, 9, 18
Q2. 72 = 2³ × 3²
Q3. 28 = 2² × 7, 42 = 2 × 3 × 7 → common: 2 × 7 = 14
Q4. 8 = 2³, 12 = 2² × 3 → LCM = 2³ × 3 = 24
Q5. 87 = 3 × 29 → three factors (1, 3, 87) → not prime
Q6. 36 = 2² × 3², 48 = 2⁴ × 3 → HCF = 2² × 3 = 12; LCM = 2⁴ × 3² = 144
Q7. LCM(8, 12) = 24 → after 24 seconds
📌 Key Facts to Memorise
- Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (2 is the only even prime)
- HCF: product of common primes at their LOWEST powers
- LCM: product of all primes at their HIGHEST powers
- Shortcut: HCF × LCM = product of the two numbers
- A factor pairs with a quotient; a multiple is the times table
🗺️ In the Exam
Primes/HCF/LCM: 2–4 marks on Paper 1. Factor trees are non-calculator material; the applied LCM contexts (buses, flashing lights) are the AO2 layer.
Links to: four operations (N2), index laws (N7), squares and roots (N6).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Memorise the first 10 prime numbers for quick recall
- Use factor trees for prime factorisation - always check your answer by multiplying
- For HCF use common factors with LOWEST powers
- For LCM use ALL factors with HIGHEST powers
- Check: HCF must be smaller than or equal to both numbers, LCM must be larger than or equal to both numbers
🧠 Problem-Solving Strategies
Problem-Solving
For HCF/LCM word problems, identify whether the question asks about a "largest" shared quantity (HCF) or a "smallest" common time/event (LCM). Use prime factorisation for reliability — list methods miss factors. When checking, verify: HCF must divide into both numbers, and LCM must be divisible by both numbers.
Multi-Step Problem
Buses leave a station every 12 minutes and trains leave every 18 minutes. If a bus and a train both leave at 9:00 am, when is the next time they both leave together?
Solution: Find LCM of 12 and 18. 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36 minutes. Next time = 9:00 + 36 min = 9:36 am.
⚠️ Common Errors
Watch Out!
1. Wrong: HCF uses highest powers of all prime factors Correct: HCF uses common prime factors with their LOWEST powers only
2. Wrong: LCM uses lowest powers of common prime factors Correct: LCM uses ALL prime factors with their HIGHEST powers
3. Wrong: 1 is a prime number Correct: 1 is NOT prime — it has only one factor, not two
✍️ 6-Mark Exam Question
Extended Answer
6 marks: (a) Express 120 and 180 as products of their prime factors. (b) Find the HCF and LCM of 120 and 180. (c) Two lighthouses flash every 120 seconds and 180 seconds. They flash together at midnight. How long will it be until they next flash together, and at what time will this be?
(a) 120 = 2³ × 3 × 5, 180 = 2² × 3² × 5
(b) HCF = 2² × 3 × 5 = 60. LCM = 2³ × 3² × 5 = 360.
(c) They flash together every LCM seconds = 360 seconds = 6 minutes. Time = 00:06 am.
Mark scheme: 2 marks for correct prime factorisations, 2 marks for HCF and LCM, 2 marks for converting to minutes and giving the time
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A gardener has two rectangular plots. Plot A is 18 m by 24 m. Plot B is 30 m by 42 m. He wants to divide both plots into equal-sized square sections with no land wasted.
(a) What is the largest possible side length of the square sections for Plot A?
(b) He wants the same square size to work for both plots. Is this possible? Explain why.
(c) What is the largest square size that would work for both plots without wasting land?
Answers: (a) HCF of 18 and 24. 18 = 2 × 3², 24 = 2³ × 3. HCF = 2 × 3 = 6 m. (b) For Plot B, HCF of 30 and 42 = 6. So 6 m works for both! (c) The largest square is the HCF of all four numbers. Check: 6 divides 18, 24, 30, and 42. Yes — 6 m squares work for both plots.
📝 Exam Questions by Topic
🎬 Video Resources
Ready to ace your GCSE Mathematics exams?
Get the best revision books and guides to boost your grades.