Factors & Multiples
Primes, factors and multiples, prime factorisation, and the HCF and LCM of two or more numbers.
Learn
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.
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.