Every whole number greater than 1 can be written as a product of prime factors, for example 90 = 2 × 3² × 5. From the prime factors, the highest common factor (HCF) is the product of the primes both numbers share, using the lowest powers. The lowest common multiple (LCM) uses every prime that appears in either number, with the highest powers.
HCF and LCM from prime factors
- Product of primes: split with a factor tree until every branch is prime
- HCF: multiply the shared primes, using the lowest power of each
- LCM: multiply every prime that appears, using the highest power of each
- HCF × LCM = the product of the two numbers
Writing a number as a product of primes
Use a factor tree: split the number into any two factors, then keep splitting until every branch ends in a prime. Collect the primes and write them in order, using index form for repeats. 90 splits into 9 × 10, then 3 × 3 × 2 × 5, so 90 = 2 × 3² × 5. Whichever factors you start with, you always end with the same primes.
Finding the HCF
The highest common factor is the largest number that divides into both. Write each number as a product of primes and multiply the primes they share, taking the lowest power of each. For small numbers, listing the factors also works, but prime factors are quicker for larger numbers.
Finding the LCM
The lowest common multiple is the smallest number that both divide into. Take every prime that appears in either number, with the highest power of each, and multiply them. For 10 = 2 × 5 and 14 = 2 × 7, the LCM is 2 × 5 × 7 = 70.
Using a Venn diagram
Put the prime factors of each number in its own circle, with shared factors in the overlap. The HCF is the product of the numbers in the overlap; the LCM is the product of everything in the diagram. This is a tidy way to organise larger numbers.
Worded problems: which one do I need?
If a question asks for the largest equal pieces or groups, that is the HCF, because it must divide into both amounts. If it asks when two repeating events next happen together, that is the LCM, because it must be a multiple of both intervals.
Worked examples
Example 1. Write 90 as a product of its prime factors.
- 90 = 9 × 10.
- 9 = 3 × 3 and 10 = 2 × 5.
Answer: 90 = 2 × 3² × 5.
Example 2. Find the HCF of 18 and 30.
- 18 = 2 × 3² and 30 = 2 × 3 × 5.
- Shared primes, lowest powers: 2 and 3.
Answer: HCF = 6.
Example 3. Find the LCM of 10 and 14.
- 10 = 2 × 5 and 14 = 2 × 7.
- Every prime, highest powers: 2 × 5 × 7.
Answer: LCM = 70.
Example 4. A = 2³ × 3 × 5 and B = 2² × 5 × 7. Find the HCF and LCM of A and B.
- HCF: shared primes 2 and 5, lowest powers: 2² × 5.
- LCM: all primes, highest powers: 2³ × 3 × 5 × 7.
Answer: HCF = 2² × 5 = 20; LCM = 2³ × 3 × 5 × 7 = 840.
Example 5. Two buses leave a station together. One returns every 14 minutes and the other every 21 minutes. After how many minutes are they next at the station together?
- This needs a common multiple of 14 and 21: the LCM.
- 14 = 2 × 7 and 21 = 3 × 7, so the LCM = 2 × 3 × 7.
Answer: 42 minutes.
Common mistakes
- Stopping the factor tree too early. Keep splitting until every branch is a prime number.
- Mixing up HCF and LCM. HCF uses shared primes with the lowest powers; LCM uses all primes with the highest powers.
- Including 1 as a prime factor. 1 is not prime, so it never appears in a prime factorisation.
- Not using index form when asked. Write 2 × 2 × 2 × 3 as 2³ × 3.
- Choosing the wrong one in a word problem. Equal pieces or groups means HCF; events happening together again means LCM.
How prime factors, HCF and LCM are examined
Prime factorisation, HCF and LCM are tested at both Foundation and Higher tier in Edexcel GCSE (1MA1), and in Edexcel International GCSE (4MA1). Typical questions ask you to:
- write a number as a product of its prime factors, often in index form
- find the HCF or LCM of two numbers
- find the HCF or LCM when the numbers are given in prime factor form
- solve a worded problem about timetables or equal groups
Try the factors, multiples and primes worksheet, which includes answers, or practise in the GCSE number revision engine.
This builds on primes, factors and multiples, and the LCM is used to find common denominators in fractions.
Frequently asked questions
What is a product of prime factors?
A number written as primes multiplied together, such as 60 = 2² × 3 × 5.
How do you find the HCF using prime factors?
Multiply the prime factors the two numbers share, taking the lowest power of each.
How do you find the LCM using prime factors?
Multiply every prime factor that appears in either number, taking the highest power of each.