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Break any number into its prime building blocks and find HCF and LCM using the Venn diagram method.
Prime factorisation is expressing a number as a product of its prime factors (e.g. 60 = 2² × 3 × 5). The Highest Common Factor (HCF) is found by multiplying the shared prime factors, while the Lowest Common Multiple (LCM) uses all prime factors at their highest powers. The Venn diagram method makes both calculations visual and reliable.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
Prime factorisation means breaking a number down into the prime numbers that multiply together to give the original number. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13...).
Start with your number and split it into any pair of factors. Keep splitting each branch until every end point is prime. Then collect all the primes and write them using index notation.
Example: Find the prime factorisation of 180
180 = 2 × 90
90 = 2 × 45
45 = 3 × 15
15 = 3 × 5
180 = 2² × 3² × 5
Alternatively, repeatedly divide by the smallest possible prime. Divide 180 by 2 to get 90, by 2 again to get 45, by 3 to get 15, by 3 to get 5. The prime factorisation is the product of all the divisors: 2² × 3² × 5.
The Venn diagram method is the gold standard for IGCSE exams because it is systematic, visual, and less prone to error than listing factors.
Check: HCF × LCM should equal the product of the two original numbers. If it does not, recheck your diagram.
Find the HCF and LCM of 24 and 36.
Step 1: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3
Step 2: Shared primes (overlap): 2, 2, 3
Step 3: Unique to 24: 2 | Unique to 36: 3
Step 4: HCF = 2 × 2 × 3 = 12
Step 5: LCM = 2 × 2 × 2 × 3 × 3 = 72
Check: 12 × 72 = 864 = 24 × 36 ✓
Find the HCF and LCM of 12, 18 and 30.
Step 1: 12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5
Step 2: Common to ALL three: 2 × 3 = 6 → HCF = 6
Step 3: For LCM, take the highest power of each prime: 2² × 3² × 5
LCM = 4 × 9 × 5 = 180
Two buses leave a station at 9:00 am. Bus A returns every 24 minutes and Bus B returns every 36 minutes. At what time will both buses next be at the station together?
Method: Find the LCM of 24 and 36 (from example above, LCM = 72).
Interpretation: 72 minutes after 9:00 am = 1 hour 12 minutes later.
Answer: 10:12 am
Mixing up HCF and LCM: HCF is the overlap (smaller number), LCM is everything (larger number). If your HCF is bigger than your LCM, you have swapped them.
Not using individual primes in the Venn diagram: Place each prime separately (2, 2, 3) not as a product (12). This avoids miscounting.
Including 1 as a prime factor: 1 is NOT a prime number. Never include it in your factor tree.
Forgetting index notation: The question may require your answer in index form (2² × 3² × 5), not as a product list.
Always draw the Venn diagram: Even if you think you can do it in your head, the diagram earns method marks and reduces errors.
Use the HCF × LCM check: Multiply your HCF by your LCM. It should equal the product of the original two numbers.
Read the question carefully: Some questions give the prime factorisation and ask you to work backwards to find the original number.
Know your primes to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Instant recognition saves valuable time on Paper 1.
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