Prime Factors, HCF & LCMEdexcel IGCSE Maths: Revision notes

Section 1

What is prime factorisation and how do I find it?

Every integer greater than 1 can be written as a unique product of prime numbers — this is called its prime factorisation.

To find it, repeatedly divide by the smallest prime that goes in exactly, until you reach 1:

$360 \div 2 = 180$
$180 \div 2 = 90$
$90 \div 2 = 45$
$45 \div 3 = 15$
$15 \div 3 = 5$
$5 \div 5 = 1$

So $360 = 2^{3} \times 3^{2} \times 5$ .

A factor tree does the same job visually: split the number into any two factors, then keep splitting each branch until every end is prime.

Key termsprime numberprime factorisationfactor tree

Exam tip

Always write the final answer using index (power) notation, e.g. $2^{3} \times 3^{2} \times 5$ , not a long list of repeated primes.

Common mistake

Don't stop dividing too early — check every branch ends in a prime, not just a small number like 4 or 9.

Section 2

How do I find the HCF using prime factors?

The Highest Common Factor (HCF) of two numbers is the largest number that divides into both exactly.

Method using prime factorisation:

Write each number as a product of primes.
For every prime that appears in both lists, take the lowest power.
Multiply these together.

Example: Find the HCF of 360 and 252.

Number	Prime factorisation
360	$2^{3} \times 3^{2} \times 5$
252	$2^{2} \times 3^{2} \times 7$

Common primes: 2 (lowest power $2^{2}$ ) and 3 (lowest power $3^{2}$ ).

$\text{HCF} = 2^{2} \times 3^{2} = 36$

Key termsHighest Common Factor (HCF)

Example

HCF of 360 and 252 is $2^{2} \times 3^{2} = 36$ .

Section 3

How do I find the LCM using prime factors?

The Lowest Common Multiple (LCM) is the smallest number that both original numbers divide into exactly.

Method using prime factorisation:

Write each number as a product of primes.
For every prime that appears in either list, take the highest power.
Multiply these together.

Example: Find the LCM of 360 and 252 (using the factorisations above).

Primes involved: 2, 3, 5, 7 — take highest powers: $2^{3}$ , $3^{2}$ , $5^{1}$ , $7^{1}$ .

$\text{LCM} = 2^{3} \times 3^{2} \times 5 \times 7 = 2520$

Key termsLowest Common Multiple (LCM)

Think of it like this

Think of HCF as sharing what's common (lowest power kept), and LCM as covering everything (highest power kept).

Section 4

How are HCF and LCM linked, and where do exam questions use them?

For any two positive integers $a$ and $b$ :

$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$

This is a useful check — for 360 and 252: $36 \times 2520 = 90{,}720$ , and $360 \times 252 = 90{,}720$ . It matches.

Common exam contexts:

HCF: sharing items into the largest equal groups (e.g. "What is the largest number of identical bags Amir can make from 360 sweets and 252 chocolates with none left over?")
LCM: events that repeat at different intervals meeting up again (e.g. "Two buses leave at the same time, one every 12 minutes and one every 18 minutes — when do they next leave together?")

Key termsproduct of HCF and LCM

Exam tip

"Largest/greatest that fits/shares equally" signals HCF; "when will they next coincide/repeat together" signals LCM.

Must Know

Every integer > 1 has a unique prime factorisation — write answers with index notation.
HCF: multiply common primes using the lowest shared power.
LCM: multiply all primes involved using the highest power seen.
Check your work: $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$ .
"Sharing equally / largest group" → HCF; "repeats together / next coincide" → LCM.
A number that is prime has itself as its only prime factor (e.g. $7 = 7^{1}$ ).

That's the notes covered.

Carry on to the next subtopic.