Factors , Multiples and Primes

Three words sit at the heart of this topic. A factor divides a number 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 9 the big skill is prime factorisation — breaking a number fully into primes — and using those primes to find the HCF and LCM far faster than listing ever could. You also stretch to numbers large enough that listing factors would be hopeless.

The primes worth knowing by heart are $2, 3, 5, 7, 11, 13, 17, 19, 23$. 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 $540 = 2 \times 2 \times 3 \times 3 \times 3 \times 5$. A string of repeated primes is hard to read, so index form tidies it: count how many times each prime appears and write it as a power.

So $540 = 2^2 \times 3^3 \times 5$. It does not matter how you start the tree — the primes are always the same. This compact form is exactly what makes the HCF and LCM quick.

The highest common factor (HCF) is the biggest number that divides into both numbers. The lowest common multiple (LCM) is the smallest number both divide into. Prime factorisation finds them quickly.

Write both numbers in index form, then:

• HCF — for each prime that appears in both, take the lower power, then multiply.
• LCM — for each prime in either number, take the higher power, then multiply.

Take $90 = 2 \times 3^2 \times 5$ and $126 = 2 \times 3^2 \times 7$. The shared primes are 2 (lower power $2^1$) and 3 (lower power $3^2$). So $\text{HCF} = 2 \times 3^2 = 18$. For the LCM, take the highest power of each prime that appears: $2 \times 3^2 \times 5 \times 7 = 630$.

A neat check: $\text{HCF} \times \text{LCM}$ equals the two numbers multiplied. Here $18 \times 630 = 11\,340$, and $90 \times 126 = 11\,340$ too.

The prime-factor method scales smoothly to three numbers — exactly the same rules apply.

Find the HCF and LCM of 24, 36 and 60. First, prime factorise each:

• $24 = 2^3 \times 3$
• $36 = 2^2 \times 3^2$
• $60 = 2^2 \times 3 \times 5$

For the HCF, a prime must appear in all three. The prime 2 appears in all three (lowest power $2^2$); 3 appears in all three (lowest power $3^1$); 5 does not. So $\text{HCF} = 2^2 \times 3 = 12$.

For the LCM, take the highest power of every prime that appears anywhere: $2^3$, $3^2$ and $5$. So $\text{LCM} = 2^3 \times 3^2 \times 5 = 360$.

Note that the simple $\text{HCF} \times \text{LCM}$ check works only for two numbers — with three, just trust the prime-factor method.

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

An HCF question is usually about splitting things into the largest equal groups. Example: 84 apples and 126 oranges are packed into identical baskets with nothing left over. The largest number of baskets is the HCF of 84 and 126. Since $84 = 2^2 \times 3 \times 7$ and $126 = 2 \times 3^2 \times 7$, the HCF is $2 \times 3 \times 7 = 42$.

An LCM question is usually about events that repeat and meet again. Example: one bus leaves every 15 minutes and another every 25 minutes. They leave together again after the LCM of 15 and 25. Since $15 = 3 \times 5$ and $25 = 5^2$, the LCM is $3 \times 5^2 = 75$ minutes.

Read each question slowly and ask: am I splitting into equal groups, or waiting for 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.