HCF, LCM and Prime Factorisation
The Skill
Prime Numbers
A prime number has exactly two factors: 1 and itself.
First ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Note: 1 is NOT a prime number (it only has one factor).
Prime Factorisation
Writing a number as a product of its prime factors.
Method: Factor Tree
60
/ \
6 10
/ \ / \
2 3 2 5
$$60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$$
Method: Repeated Division $$60 \div 2 = 30$$ $$30 \div 2 = 15$$ $$15 \div 3 = 5$$ $$5 \div 5 = 1$$ So $60 = 2^2 \times 3 \times 5$
HCF — Highest Common Factor
The largest number that divides into both numbers exactly.
Using prime factorisation: Multiply the common prime factors (use the lower power).
$$24 = 2^3 \times 3$$ $$36 = 2^2 \times 3^2$$
Common factors: $2^2$ and $3^1$ $$\text{HCF} = 2^2 \times 3 = 12$$
LCM — Lowest Common Multiple
The smallest number that both numbers divide into exactly.
Using prime factorisation: Multiply all prime factors (use the higher power).
$$24 = 2^3 \times 3$$ $$36 = 2^2 \times 3^2$$
All factors with higher powers: $2^3$ and $3^2$ $$\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72$$
Quick Check
$$\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$$ $$12 \times 72 = 864 = 24 \times 36$$ ✓
The Traps
Common misconceptions and how to avoid them.
Confusing HCF and LCM "The HCF-LCM Swap"
The Mistake in Action
Find the HCF of 12 and 18.
Wrong: $12 = 2^2 \times 3$ $18 = 2 \times 3^2$ HCF = $2^2 \times 3^2 = 36$
Why It Happens
Students mix up whether to use the higher or lower powers, or confuse which one is "highest" and which is "lowest".
The Fix
HCF = Highest Common Factor
- Use only the factors that appear in both numbers
- Take the lower power of each
- HCF is always smaller than or equal to both numbers
LCM = Lowest Common Multiple
- Use all factors from both numbers
- Take the higher power of each
- LCM is always larger than or equal to both numbers
For 12 and 18:
- Common factors: 2 and 3
- Lower powers: $2^1$ and $3^1$
- HCF = $2 \times 3 = 6$
Spot the Mistake
Find the HCF of 12 and 18
$12 = 2^2 \times 3$, $18 = 2 \times 3^2$
HCF = $2^2 \times 3^2 = 36$
Click on the line that contains the error.
Finding LCM by just multiplying the two numbers "The Multiply Mistake"
The Mistake in Action
Find the LCM of 6 and 8.
Wrong: $6 \times 8 = 48$
Why It Happens
Students think "lowest common multiple" means the first multiple they can find, so they just multiply the numbers together.
The Fix
Multiplying gives a common multiple, but not necessarily the lowest.
$6 \times 8 = 48$ is a common multiple, but is it the lowest?
Check:
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...
- Multiples of 8: 8, 16, 24, 32, 40, 48...
The lowest common multiple is 24, not 48.
Proper method: $6 = 2 \times 3$ $8 = 2^3$ LCM = $2^3 \times 3 = 24$
Spot the Mistake
Find the LCM of 6 and 8
$6 \times 8 = 48$
Click on the line that contains the error.
Thinking 1 is a prime number "The One Error"
The Mistake in Action
List all the prime factors of 30.
Wrong: 1, 2, 3, 5
Why It Happens
Students remember that primes "only divide by 1 and themselves" and incorrectly include 1.
The Fix
A prime number has exactly two factors: 1 and itself.
The number 1 only has one factor (itself), so it is NOT prime.
Prime factors of 30: 2, 3, 5 (not 1!)
$$30 = 2 \times 3 \times 5$$
Key primes to know: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Spot the Mistake
List all prime factors of 30
1, 2, 3, 5
Click on the line that contains the error.
Writing prime factorisation without index notation "The Index Omission"
The Mistake in Action
Write 72 as a product of prime factors.
Wrong: $72 = 2 \times 2 \times 2 \times 3 \times 3$
Why It Happens
Students correctly find all the prime factors but don't use index notation when the question requires it.
The Fix
When asked for a product of prime factors, use index notation unless told otherwise.
$$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$$
Index notation is:
- Required in most exam questions
- Neater and clearer
- Essential for finding HCF and LCM efficiently
Always check: "Does my answer use powers?"
Spot the Mistake
Write 72 as a product of prime factors
$72 = 2 \times 2 \times 2 \times 3 \times 3$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Write 180 as a product of prime factors. Give your answer in index form.
Solution
Method: Factor Tree
180
/ \
18 10
/ \ / \
2 9 2 5
/ \
3 3
Reading the prime factors from the tree: $$180 = 2 \times 2 \times 3 \times 3 \times 5$$
In index form: $$180 = 2^2 \times 3^2 \times 5$$
Answer: $2^2 \times 3^2 \times 5$
Question
Write 420 as a product of prime factors.
Solution
Method: Repeated Division
Divide by the smallest prime that works, repeatedly:
$420 \div 2 = 210$ $210 \div 2 = 105$ $105 \div 3 = 35$ $35 \div 5 = 7$ $7 \div 7 = 1$
Reading down the left side: $$420 = 2 \times 2 \times 3 \times 5 \times 7$$ $$420 = 2^2 \times 3 \times 5 \times 7$$
Answer: $2^2 \times 3 \times 5 \times 7$
Question
$A = 2^3 \times 3^2 \times 5$ and $B = 2^2 \times 3^4 \times 7$
Find (a) the HCF of A and B, (b) the LCM of A and B.
Solution
(a) HCF — common factors, lower powers
| Prime | In A | In B | Common? | Lower power |
|---|---|---|---|---|
| 2 | $2^3$ | $2^2$ | Yes | $2^2$ |
| 3 | $3^2$ | $3^4$ | Yes | $3^2$ |
| 5 | $5^1$ | — | No | — |
| 7 | — | $7^1$ | No | — |
$$\text{HCF} = 2^2 \times 3^2 = 4 \times 9 = 36$$
(b) LCM — all factors, higher powers
| Prime | In A | In B | Higher power |
|---|---|---|---|
| 2 | $2^3$ | $2^2$ | $2^3$ |
| 3 | $3^2$ | $3^4$ | $3^4$ |
| 5 | $5^1$ | — | $5^1$ |
| 7 | — | $7^1$ | $7^1$ |
$$\text{LCM} = 2^3 \times 3^4 \times 5 \times 7 = 8 \times 81 \times 5 \times 7 = 22680$$
Answers: (a) 36, (b) 22680
Level 2: Scaffolded
Fill in the key steps.
Question
Find the HCF of 48 and 60.
Question
Find the LCM of 48 and 60.
Level 3: Solo
Try it yourself!
Question
Two lighthouses flash their lights. Lighthouse A flashes every 8 seconds. Lighthouse B flashes every 12 seconds. They flash together at midnight. After how many seconds will they next flash together?
Show Solution
We need to find when both lighthouses flash at the same time.
This is the LCM of 8 and 12.
Prime factorisations: $8 = 2^3$ $12 = 2^2 \times 3$
LCM (all primes, higher powers): $\text{LCM} = 2^3 \times 3 = 8 \times 3 = 24$
Answer: 24 seconds
Check:
- In 24 seconds, A flashes $24 \div 8 = 3$ times ✓
- In 24 seconds, B flashes $24 \div 12 = 2$ times ✓
Examiner's View
Mark allocation: Prime factorisation: 2 marks. HCF or LCM: 2-3 marks. Both together: 4 marks.
Common errors examiners see:
- Including 1 or the number itself as prime factors
- Confusing HCF and LCM
- Not using index notation when required
- Errors in the factor tree
What gains marks:
- Using index notation ($2^3$ not $2 \times 2 \times 2$)
- Showing the factor tree or division method
- Clearly identifying common/all factors for HCF/LCM
AQA Notes
AQA often asks for answers in index form. Make sure you use powers!