Identify factors, multiples and primes, write a number as a product of prime factors in index form, and use prime factorisation to find the highest common factor and lowest common multiple.

What this dot point is asking

WJEC requires you to identify factors, multiples and prime numbers, to write any number as a product of its prime factors in index form, and to use that prime factorisation to find the highest common factor (HCF) and lowest common multiple (LCM) of two or more numbers. These appear on both components and underpin simplifying fractions, working with surds and solving timing problems, so the prime-factor method is a high-value technique to make automatic.

Factors, multiples and primes

A factor of a number divides it with no remainder, so the factors of $12$ are $1, 2, 3, 4, 6, 12$. A multiple is the result of multiplying by a whole number, so multiples of $12$ are $12, 24, 36, \ldots$ Factors come in pairs that multiply to the number, which is why finding factors in pairs ($1 \times 12$, $2 \times 6$, $3 \times 4$) guarantees you find them all.

Writing a number as a product of primes

Every whole number greater than $1$ has a unique prime factorisation. A factor tree is the standard tool: split the number into any factor pair, keep splitting composite factors, and circle the primes at the ends.

The answer is the same whichever factor pair you start with, because the prime factorisation is unique.

Finding the HCF and LCM

Once both numbers are in prime-factor form, the HCF and LCM follow from simple rules.

For $36 = 2^2 \times 3^2$ and $60 = 2^2 \times 3 \times 5$: shared primes are $2$ and $3$, so the HCF is $2^2 \times 3 = 12$; all primes are $2, 3, 5$ at their highest powers, so the LCM is $2^2 \times 3^2 \times 5 = 180$.

A Venn diagram makes this visual: place shared primes in the overlap and the rest in the outer regions. The overlap multiplies to the HCF and the whole diagram multiplies to the LCM.

Where HCF and LCM problems hide

Worded questions rarely use the words HCF or LCM. An HCF problem asks for the largest equal group, for example the largest number of identical party bags you can make from $84$ sweets and $120$ stickers. An LCM problem asks when repeating events coincide, for example two buses leaving every $18$ and $24$ minutes meeting again after the LCM of $18$ and $24$, which is $72$ minutes.

Why this matters

Prime factorisation is the engine behind several other topics: it simplifies fractions to lowest terms, it underlies simplifying surds by spotting square factors, and it solves real timing and grouping problems through the HCF and LCM. Because WJEC sets these on the non-calculator Unit 1, a quick, reliable factor-tree method protects marks and speeds up everything that depends on it.