Factors, Multiples and Primes

Three familiar words sit at the heart of this topic.

A factor of a number divides into it exactly, with no remainder. A multiple is what you get by multiplying a number by a whole number. A prime number has exactly two factors — 1 and itself.

At Stage 8 you put these ideas to work. The big skill is prime factorisation — breaking a number fully into primes — and then using those primes to find the HCF and LCM of larger numbers far faster than listing ever could.

The primes worth knowing by heart are $2, 3, 5, 7, 11, 13, 17, 19$. They are the building blocks for everything in this guide.

Every whole number above 1 can be written as a product of primes — and there is only one way to do it. The friendliest method is a factor tree: split the number into any two factors, then keep splitting until every branch ends on a prime.

The green leaves give $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$. That is correct, but a string of repeated primes is hard to read. Index form tidies it: count how many times each prime appears and write it as a power.

So $360 = 2^3 \times 3^2 \times 5$. This compact form is exactly what makes the HCF and LCM easy.

The highest common factor (HCF) of two numbers is the biggest number that divides into both. For small numbers, listing factors works — but for larger numbers, prime factorisation is far faster.

The method has three steps:

1. Write both numbers in index form.
2. For each prime, take the lower power (or skip the prime if it is not in both).
3. Multiply those shared pieces together.

Take $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$. The prime 2 appears in both — the lower power is $2^3$. The prime 3 appears in both — the lower power is $3^1$. The prime 5 is only in 120, so it is skipped.

So $\text{HCF} = 2^3 \times 3 = 24$. The HCF is the biggest factor 72 and 120 share.

The lowest common multiple (LCM) of two numbers is the smallest number that both divide into. Prime factorisation makes it just as quick as the HCF.

The method:

1. Write both numbers in index form.
2. For each prime that appears in either number, take the highest power.
3. Multiply all of those together.

For $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$: the highest power of 2 is $2^3$, of 3 is $3^2$, of 5 is $5^1$.

So $\text{LCM} = 2^3 \times 3^2 \times 5 = 360$. A neat check: $\text{HCF} \times \text{LCM}$ equals the product of the two numbers. Here $24 \times 360 = 8640$, and $72 \times 120 = 8640$ too.

The real power of HCF and LCM shows in word problems. The trick is recognising which one a question is really asking for.

An HCF question is usually about splitting things into the largest equal groups. Example: 72 pens and 120 pencils are packed into identical kits with nothing left over. The largest number of kits is the HCF of 72 and 120, which is 24.

An LCM question is usually about events that repeat and meet again. Example: one light flashes every 12 seconds and another every 18 seconds. They flash together again after the LCM of 12 and 18. Since $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$, the LCM is $2^2 \times 3^2 = 36$ seconds.

Read the question slowly and ask: am I splitting into equal groups, or waiting for two cycles to line up?

These ideas reach far beyond this topic:

• Fractions lean on the HCF to simplify, and on the LCM to find common denominators when adding or subtracting.
• Algebra uses factors when you factorise expressions — pulling out a common factor mirrors finding the HCF of numbers.
• Powers and roots connect here, since prime factorisation is the clearest way to write a number in index form.
• Problem solving often hides an LCM, such as working out when two repeating timetables next coincide.

If a fractions or algebra topic later feels shaky, it is often the HCF or LCM underneath that needs a quick refresh. These number-building skills quietly support a huge amount of maths to come.