Identifying factors, multiples and primes, writing a number as a product of prime factors, and finding the HCF and LCM.

What this dot point is asking

AQA wants you to identify factors, multiples and prime numbers, express a number as a product of prime factors in index form, and use prime factorisation to find the highest common factor (HCF) and lowest common multiple (LCM). These ideas underpin simplifying fractions, working with surds, and solving "when do two events coincide" problems, so they recur across the number strand.

Factors, multiples and primes

A factor of a number divides into it with no remainder; the factors of $12$ are $1, 2, 3, 4, 6, 12$. A multiple is any number in the times table; multiples of $12$ are $12, 24, 36, \ldots$ A prime number has exactly two distinct factors, $1$ and itself, so $2, 3, 5, 7, 11, 13$ are prime but $1$ is not (it has only one factor) and $9$ is not (it has three). A useful fact: $2$ is the only even prime.

Prime factorisation

The fundamental theorem of arithmetic says every integer above $1$ is either prime or a unique product of primes. A factor tree finds this product: split the number into any factor pair, keep splitting composite numbers, and stop when every branch ends in a prime.

Highest common factor and lowest common multiple

Once you have the prime factorisations, the HCF and LCM follow simple rules.

For $48 = 2^4 \times 3$ and $60 = 2^2 \times 3 \times 5$, the shared primes are $2$ and $3$. The HCF takes $2^2$ and $3^1$, giving $4 \times 3 = 12$. The LCM takes the highest powers $2^4$, $3^1$ and $5^1$, giving $16 \times 3 \times 5 = 240$. As a check, $12 \times 240 = 2880 = 48 \times 60$.

A Venn-diagram method places the 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.

Why the HCF and LCM rules work

The reasoning is worth understanding, not just memorising. A common factor must divide both numbers, so it can only contain primes present in both, and no more of each prime than the smaller number supplies, which is exactly the lowest power. The highest common factor is the largest such number, hence the lowest powers of shared primes. A common multiple must be divisible by both, so it must contain at least as many of each prime as the larger number needs, which is the highest power. The lowest common multiple is the smallest such number, hence the highest powers of all primes.

Where HCF and LCM are used

These appear in real contexts disguised as word problems. The LCM answers "when do two repeating events next coincide": two buses leaving every $12$ and $18$ minutes next leave together after the LCM, $\text{LCM}(12, 18) = 36$ minutes. The HCF answers "what is the largest equal grouping": cutting two ribbons of $48\,\text{cm}$ and $60\,\text{cm}$ into equal whole-centimetre pieces with none left over uses the HCF, $\text{HCF}(48, 60) = 12\,\text{cm}$. Recognising which rule a word problem needs (coincidence means LCM, largest equal share means HCF) is the real exam skill.