Number - Structure and calculation: use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem

The idea

A factor divides a number exactly; a multiple is found in a number's times table. The Highest Common Factor (HCF) is the largest number that is a factor of two or more numbers. The Lowest Common Multiple (LCM) is the smallest number that is a multiple of them. A prime number has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7). The unique factorisation theorem states every integer greater than 1 can be written as a unique product of prime factors. This prime factorisation can be found using a factor tree. For example, 60 = 2 × 2 × 3 × 5, written in product notation as 2² × 3 × 5. To find the HCF and LCM of two numbers like 60 and 90 (90 = 2 × 3² × 5), use their prime factors. The HCF is the product of the lowest powers of their shared primes: 2¹ × 3¹ × 5¹ = 30. The LCM is the product of the highest powers of all primes that appear in either factorisation: 2² × 3² × 5¹ = 180.

What earns the marks

• Correctly listing factors or multiples of a number.
• A complete factor tree or repeated division method showing only prime numbers at the end.
• The prime factorisation written correctly as a product, using index notation for repeated primes.
• Selecting the correct prime powers for the HCF (lowest powers of common primes) or LCM (highest powers of all primes).
• Evaluating the final product for the HCF or LCM correctly.

Common mistakes and how to put them right

• Confusing factors and multiples, e.g. listing multiples of 12 when asked for factors.

• Calling 1 a prime number, or not recognising that 2 is the only even prime.

• Stopping a factor tree at a composite branch such as 6 or 10 and treating it as a prime.

• Swapping the methods for HCF and LCM, for example by using the highest powers for the HCF.

Exam tips

• When finding both HCF and LCM from prime factorisations, a Venn diagram can help. The intersection gives the HCF, and the product of all numbers in the diagram gives the LCM.
• Multiply your prime factorisation back out as a check before you use it for anything else.
• Read the question carefully to see if the answer is wanted as a product of primes or as a single number.