Prime factors, HCF and LCM
Every whole number above 1 can be written as a product of primes in exactly one way. Once you have those prime factors, the highest common factor and the lowest common multiple of two numbers drop straight out.
Part 1 of 3: Learn it
In short
- A prime has exactly two factors, 1 and itself. 1 is not prime; 2 is the only even prime.
- Use a factor tree, then write the answer in index form: 360 = 2³ × 3² × 5.
- HCF: multiply the primes the numbers share. LCM: multiply the HCF by everything left over.
Where this is in your specification
Spec points: DfE N4
| Board | Topic: Place value, rounding and bounds |
|---|---|
| DfE content | N1, N2, N3, N4, N5, N14, N15, N16 |
| AQA 8300 | Number |
| Edexcel 1MA1 | Number |
| OCR J560 | 1 Number operations and integers; 4 Approximation and estimation |
| Eduqas C300 | HN1 to HN5, HN13 to HN16 |
| Cambridge IGCSE 0580 | E1.1, E1.2, E1.5, E1.6, E1.9, E1.10, E1.14 |
| National 5 C847 75 | Numerical skills: significant figures |
Factors, multiples and primes
- Factors of 18: 1, 2, 3, 6, 9, 18. Find them in pairs so none is missed.
- Multiples of 7: 7, 14, 21, 28, ...
- Primes below 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Prime factor decomposition
Split the number into any factor pair, then keep splitting until every branch ends in a prime. Circle the primes and multiply them back together to check. Write repeated primes as powers: 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5 = 360.
HCF and LCM
Put the prime factors of both numbers in a Venn diagram, with shared primes in the overlap. The HCF is the product of the overlap; the LCM is the product of everything in the diagram.
Word problems give it away: things happening together again (buses, lights, alarms) need the LCM; splitting into the largest equal groups needs the HCF.
Write 150 as a product of prime factors in index form.
Show the answer
150 = 2 × 3 × 5².
Part 2 of 3: See it worked
Worked examples
Example 1
Find the HCF and LCM of 84 and 120.
- 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5
- Shared: 2 × 2 × 3, so the HCF = 12
- LCM = 12 × (7 from 84) × (2 × 5 from 120) = 12 × 7 × 10
Answer: HCF = 12, LCM = 840.
Example 2
One bus leaves every 12 minutes and another every 18 minutes. Both leave at 9:00 am. When do they next leave together?
- 12 = 2² × 3 and 18 = 2 × 3²
- LCM = 2² × 3² = 36
- 36 minutes after 9:00 am
Answer: 9:36 am.
Common mistakes
- Including 1 as a prime factor.
- Stopping the factor tree at a number that is not prime, such as 9.
- Mixing up HCF and LCM in word problems.
- Multiplying the two numbers together and calling it the LCM. That is a common multiple, but often not the lowest.
Find the HCF of 48 and 72.
Show the answer
48 = 2⁴ × 3 and 72 = 2³ × 3², so the HCF = 2³ × 3 = 24.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Write 150 as a product of prime factors in index form.
150 = 2 × 3 × 5².
Find the HCF of 48 and 72.
48 = 2⁴ × 3 and 72 = 2³ × 3², so the HCF = 2³ × 3 = 24.
Find the LCM of 8 and 14.
8 = 2³ and 14 = 2 × 7, so the LCM = 2³ × 7 = 56.
Jobs that use this
- IT security co-ordinator (s'ouvre dans un nouvel onglet)
- Software developer (s'ouvre dans un nouvel onglet)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Ces fiches sont en anglais car elles suivent les programmes d'examen britanniques.
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses