Factors & Multiples
Primes, factors and multiples, prime factorisation, and the HCF and LCM of two or more numbers.
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Every point below, worked through in one video, and each point links straight to its own part.
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Factors & Multiples, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Identify factors, multiples and prime numbers
Factors divide into a number exactly (the factors of 12 are 1, 2, 3, 4, 6, 12); multiples are its times table (12, 24, 36, …). A prime has exactly two factors - 1 and itself - so 2 is the only even prime, and 1 is not prime at all.
Write a number as a product of its prime factors
Use a factor tree: split the number into any two factors and keep splitting until every branch ends in a prime. For 60: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5, written 2² × 3 × 5. Every number has exactly one prime factorisation.
Find the highest common factor (HCF) and lowest common multiple (LCM) of two numbers
The HCF is the biggest number dividing both; the LCM is the smallest number both divide into. For small numbers, list factors or multiples: HCF of 18 and 24 is 6, LCM is 72. A useful check: HCF × LCM = the product of the two numbers.
Use prime factorisations (including Venn diagrams) for HCF and LCM problems
Put each number's prime factors in a Venn diagram, sharing the common ones in the overlap. The HCF is the product of the overlap; the LCM is the product of everything in the diagram. For 60 = 2² × 3 × 5 and 72 = 2³ × 3², the HCF is 2² × 3 = 12 and the LCM is 2³ × 3² × 5 = 360.
Solve real-life problems with HCF and LCM - like when two buses next leave together
"When do they next coincide?" is an LCM question: buses every 12 and every 20 minutes next leave together after the LCM, 60 minutes. "What is the biggest equal group?" is an HCF question. Spotting which one is being asked is most of the work.
Common mistakes in Factors & Multiples
The slips that cost marks year after year, and what to do instead.
Mixing up HCF and LCM in a worded problem, such as using the HCF for "when do they next leave together".
Buses every 12 and 20 minutes next leave together after the LCM, 60 minutes. Sharing 18 pens and 24 pencils into as many identical packs as possible needs the HCF, 6.
Stopping a factor tree at a number that is not prime, as in 60 = 2 × 2 × 15.
Keep splitting until every branch ends in a prime, then write the answer as a product: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
Multiplying the two numbers to get the LCM, so the LCM of 12 and 18 becomes 216.
The product is a common multiple but not always the lowest. Go through the multiples of 18 until one divides by 12: 18 does not, 36 does, so the LCM is 36.
Multiplying every prime in the Venn diagram to find the HCF, rather than just the overlap.
The HCF uses only the shared primes, and the LCM uses them all. For 60 = 2² × 3 × 5 and 72 = 2³ × 3², the HCF is 2² × 3 = 12.
Mixing up factors and multiples, so the factors of 12 are listed as 12, 24, 36.
Factors divide into the number exactly, so they are never bigger than it: 1, 2, 3, 4, 6 and 12. Multiples are its times table: 12, 24, 36 and so on.
Practice
Try a Factors & Multiples question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Write 60 as a product of its prime factors and then evaluate 2² × 3 × 5.
- 120
- 60
- 45
- 30
Show the answer and the working
Answer: 60
Repeated division by primes shows 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
- Break down 60: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5.
- In index form this is 2² × 3 × 5.
- Evaluate: 4 × 3 × 5 = 60.