Identify factors, multiples and primes; write a number as a product of its prime factors; and use prime factorisation to find the HCF and LCM of two or more numbers.

What this dot point is asking

OCR reference N4 asks you to identify factors, multiples and primes, to express a number as a product of its prime factors, and to use that decomposition to find the highest common factor (HCF) and lowest common multiple (LCM). Prime factorisation is the engine: once a number is broken into primes, the HCF and LCM follow from simple rules about powers. The topic appears on every tier and frequently on the non-calculator paper, so the factor-tree method must be quick by hand.

Factors, multiples and primes

A factor of a number divides into it with no remainder; a multiple is what you get by multiplying it by an integer. The factors of $12$ are $1, 2, 3, 4, 6, 12$; the first few multiples of $12$ are $12, 24, 36, 48$. A prime number has exactly two distinct factors, $1$ and itself, so $2, 3, 5, 7, 11, 13$ are prime. Note that $1$ is not prime (it has only one factor) and $2$ is the only even prime.

Writing a number as a product of primes

The standard tool is a factor tree. Split the number into any factor pair, then keep splitting each composite branch until every leaf is prime. Dividing by the smallest prime first keeps the work tidy.

For $84$: $84 = 2 \times 42 = 2 \times 2 \times 21 = 2 \times 2 \times 3 \times 7$, so $84 = 2^2 \times 3 \times 7$. Writing the answer in index form, with the primes in ascending order, is the convention OCR expects and makes the HCF and LCM steps that follow much easier.

Finding the HCF and LCM from prime factors

Once both numbers are in prime-power form, two short rules give the HCF and the LCM.

A Venn diagram organises this. Place the shared prime factors in the overlap and the leftover factors in the outer regions. The overlap alone multiplies to the HCF; the entire diagram multiplies to the LCM.

Where HCF and LCM are used

HCF problems usually ask for the largest equal group: "the greatest number of identical bags you can make from $24$ red and $90$ blue beads" is the HCF, $6$. LCM problems ask when repeating events coincide: "two buses leave together; one every $24$ minutes, the other every $90$ minutes, when do they next leave together?" is the LCM, $360$ minutes. Reading the context to decide which you need is itself an AO2 reasoning skill that OCR rewards. Prime factorisation also feeds directly into simplifying surds and into index laws, so the fluency pays off beyond this single topic.