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
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).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses