1MA1 · N4NumberFoundation & HigherGrades 2–5

Factors, Multiples & Primes

Two buses leave the station together, one every 12 minutes and one every 18. When do they next leave together? A florist has 60 roses and 84 tulips and wants identical bouquets using every flower — how many bouquets? Both questions are about how numbers are built, and both are solved by breaking numbers into their prime factors.

45 min Number Fundamentals

The big picture

Every whole number greater than 1 is either prime or can be written as a product of primes in exactly one way. That product, the prime factor decomposition, is like a number’s fingerprint.

Once numbers are written as prime products, the highest common factor and lowest common multiple can be read off directly. Edexcel regularly sets these in context, so deciding which one a problem needs is as important as calculating it.

What you'll learn

Identify factors, multiples and prime numbers
Write a number as a product of prime factors using index notation
Find the HCF and LCM of two or three numbers
Decide whether a context needs the HCF or the LCM

Factors, multiples and primes

A $factor$ of $n$ divides into $n$ exactly. Factors come in pairs: the factors of $36$ are $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ and $6 \times 6$ .

A $multiple$ of $n$ is in the $n$ times table: $7, 14, 21, \dots$

A $prime$ has exactly two factors, 1 and itself. $1$ is not prime (it has only one factor) and $2$ is the only even prime. The primes under 30 are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ .

Tip — To list factors systematically, test divisors from 1 upwards and stop once the pairs meet in the middle.

Prime factor decomposition

Use a factor tree: split the number into any two factors, keep splitting until every branch ends in a prime, then multiply the primes together.

Write the answer as a product using indices, in ascending order of primes: $360 = 2^{3} \times 3^{2} \times 5$ .

Whichever split you start with, you always reach the same primes — that uniqueness is what makes the method reliable.

540 = 2 \times 270 = 2 \times 2 \times 135

135 = 3 \times 45 = 3 \times 3 \times 15 = 3 \times 3 \times 3 \times 5

3Collect with indices.

Answer

540 = 2^{2} \times 3^{3} \times 5

Your turnSame method, new numbers — try it before moving on.

Write 360 as a product of its prime factors, in index form.

Your answer

Type it like 2^3 x 3^2 x 5 (ascending order).

Why this works

Prime factors reveal properties at a glance. A number is a perfect square exactly when every prime appears to an even power — so $2^{2} \times 3^{3} \times 5$ is not a square, but multiplying by $3 \times 5$ would make it one.

HCF and LCM

The $highest common factor$ (HCF) is the largest number dividing both. Take each prime the numbers $share$ , to the $lower$ power.

The $lowest common multiple$ (LCM) is the smallest number both divide into. Take $every$ prime that appears, to the $higher$ power.

A Venn diagram of the prime factors shows both at once: the overlap multiplies to the HCF, and everything in the diagram multiplies to the LCM.

84 = 2^{2} \times 3 \times 7

and

120 = 2^{3} \times 3 \times 5

2HCF: shared primes 2 and 3, lower powers:

2^{2} \times 3 = 12

3LCM: all primes, higher powers:

2^{3} \times 3 \times 5 \times 7 = 840

AnswerHCF

= 12

, LCM

= 840

Your turnSame method, new numbers — try it before moving on.

Find the highest common factor of 72 and 90.

Your answer

HCF or LCM in context?

If the question is about $splitting$ things into the largest equal groups, it is an HCF problem.

If it is about events $repeating$ and coinciding, or the smallest amount that works for several group sizes, it is an LCM problem.

1Coinciding repeats: LCM.

12 = 2^{2} \times 3

18 = 2 \times 3^{2}

; LCM

= 2^{2} \times 3^{2} = 36

AnswerAt 9:36

Your turnSame method, new numbers — try it before moving on.

Two lights flash together. One flashes every 15 seconds, the other every 25 seconds. After how many seconds do they next flash together?

Your answer

Tip — Finish context questions with a sentence that answers the actual question — a time, a number of bouquets — not just "LCM = 36".

Think like an examiner

Write prime factor decompositions with index notation and multiplication signs, not a list of primes.
Show the decompositions used to find the HCF or LCM.
State clearly whether the context needs the HCF or LCM.

Factors and multiples

HCF: shared primes, lowest powers

Highest common factor.

LCM: all primes, highest powers

Lowest common multiple.

a \times b = HCF \times LCM

For two positive integers.

Watch out for these

Calling 1 a prime number.

A prime has exactly two factors; 1 has only one.

Giving the answer to a prime factor question as "2, 2, 3, 3, 3, 5".

Write it as a product:

2^{2} \times 3^{3} \times 5

Using the highest powers for the HCF.

HCF uses the lowest powers of shared primes.

Using the HCF for "when will they next coincide?".

Coinciding events need the LCM.

Stretch yourself

The HCF of two numbers is $18$ and their LCM is $540$ . One of the numbers is $108$ . Find the other, and explain how you know it is correct.

Hint — For two positive integers, the product of the numbers equals the product of the HCF and LCM.

Questions students ask

Why is 2 the only even prime?

Every other even number has 2 as a factor, so it has at least three factors.

Can I find the HCF by listing factors?

Yes for small numbers, but prime factors are faster and more reliable for larger ones and are expected at GCSE.

How do I find the HCF of three numbers?

Take primes shared by all three, to the lowest power that appears in any of them.

Key takeaways

Primes have exactly two factors; 1 is not prime.
Every integer above 1 has a unique prime factor decomposition.
HCF: shared primes to lowest powers. LCM: all primes to highest powers.
Splitting into equal groups needs HCF; repeating events need LCM.
For two numbers, product $=$ HCF $\times$ LCM.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

Write 1260 as a product of its prime factors.[2]

Question 2Grade 42 marks

$A = 2^{3} \times 3 \times 5^{2}$ and $B = 2 \times 3^{3} \times 5$ .

(a)Find the HCF of $A$ and $B$ .[1]
(b)Find the LCM of $A$ and $B$ .[1]

Question 3Grade 53 marks

Pencils are sold in packs of 24. Rubbers are sold in packs of 40. Sam buys the same number of pencils and rubbers.

What is the smallest number of packs of pencils Sam can buy?[3]

Unlimited practice

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Factors, Multiples & Primes

Factors, multiples and primes

Prime factor decomposition

HCF and LCM

HCF or LCM in context?

Practise it the way it is examined

Skill drill

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Related topics

Factors, Multiples & Primes

Factors, multiples and primes

Prime factor decomposition

HCF and LCM

HCF or LCM in context?

Practise it the way it is examined

Skill drill

Practise this topic

Related topics