N4NumberFoundation & HigherGrades 2–5

Factors, Multiples & Primes

Factors divide into a number exactly; multiples are the number's times table; primes have exactly two factors. Every whole number greater than 1 can be built from primes in exactly one way, and that single fact gives a fast, reliable method for the highest common factor (HCF) and lowest common multiple (LCM).

40 min Video by Maths Genie AQA GCSE Maths

The big picture

Primes are the building blocks of multiplication, the way atoms are the building blocks of matter. Writing $360 = 2^{3} \times 3^{2} \times 5$ tells you everything about 360's factors at a glance.

HCF and LCM questions on AQA papers are often disguised as problems: tiles fitting exactly, buses leaving together, packs of items that must match up. Spotting "largest that divides both" (HCF) versus "first time they coincide" (LCM) is half the marks.

Factors, Multiples and PrimesCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Identify factors, multiples and prime numbers
Write a number as a product of prime factors in index form
Find the HCF and LCM of two numbers
Solve problems involving HCF and LCM

Factors, multiples and primes

A $factor$ of $n$ divides $n$ exactly. Find them in pairs: $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ , $6 \times 6$ .

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

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

1(a) 17 and 29 have only two factors.

2(b)

15, 21, 24, 27

are in the 3 times table.

3(c)

48 \div 24 = 2

exactly.

Answer(a)

17, 29

(b)

15, 21, 24, 27

(c)

24

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

How many factors does 36 have?

Your answer

Tip — Square numbers are the only numbers with an odd number of factors — one factor pairs with itself.

Prime factorisation

Use a factor tree: split the number into any factor pair, and keep splitting until every branch ends in a prime. Circle the primes.

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

Circled primes:

2, 2, 2, 3, 3, 5

, so

360 = 2^{3} \times 3^{2} \times 5

84 = 2 \times 42 = 2 \times 2 \times 21 = 2 \times 2 \times 3 \times 7

Answer

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

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

Write 150 as a product of its prime factors in index form.

Your answer

Type it like 2^2 x 3 x 7 (ascending order).

HCF and LCM

The $HCF$ is the largest number that divides both. The $LCM$ is the smallest number that both divide into.

Using prime factors: the HCF multiplies the primes the numbers $share$ (lowest power of each shared prime). The LCM multiplies $every$ prime that appears (highest power of each). A Venn diagram shows this neatly.

60 = 2^{2} \times 3 \times 5

and

72 = 2^{3} \times 3^{2}

. HCF = middle

= 2 \times 2 \times 3 = 12

. LCM = everything

= 5 \times 2 \times 2 \times 3 \times 2 \times 3 = 360

60 = 2^{2} \times 3 \times 5

;

72 = 2^{3} \times 3^{2}

2HCF: shared primes, lowest powers:

2^{2} \times 3 = 12

3LCM: all primes, highest powers:

2^{3} \times 3^{2} \times 5 = 360

AnswerHCF

12

, LCM

360

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

Find the LCM of 18 and 30.

Your answer

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

A room measures 240 cm by 390 cm. It is covered with square tiles, the largest possible size that fit exactly with no cutting. What is the side length of a tile, in cm?

Your answer

A hidden check

For any two numbers, $HCF \times LCM =$ the product of the numbers: $12 \times 360 = 4320 = 60 \times 72$ . The shared primes are counted once in each, and every prime is counted exactly twice overall.

Think like an examiner

"Give your answer in index form" means powers are required: $2 \times 2 \times 2 \times 3$ loses a mark; $2^{3} \times 3$ scores.
Show your prime factor trees or lists — method marks are available even if the final multiplication slips.
In problem questions, write whether you are finding the HCF or LCM and why.

Key ideas

360 = 2^{3} \times 3^{2} \times 5

Prime factorisation in index form.

HCF: shared primes, lowest powers

Largest number dividing both.

LCM: all primes, highest powers

Smallest number both divide into.

Watch out for these

Counting 1 as a prime number.

1 has only one factor, so it is not prime.

Giving the product of primes as

2 + 2 + 3 + 7

It is a product:

2^{2} \times 3 \times 7

Mixing up HCF and LCM: the HCF of 12 and 20 is 60.

HCF is at most the smaller number: 4. LCM is 60.

Stopping a factor tree at a non-prime:

84 = 4 \times 21

Keep going:

2 \times 2 \times 3 \times 7

Stretch yourself

$A = 2^{3} \times 3 \times 5^{2}$ and $B = 2 \times 3^{2} \times 5$ . (a) Find the HCF of $A$ and $B$ . (b) Find the smallest whole number $k$ such that $A k$ is a square number.

Hint — A square number has every prime raised to an even power.

Questions students ask

Is there a quick way to test if a number is prime?

Try dividing by each prime up to its square root. For 97, test 2, 3, 5 and 7 (

97 < 10

). None divide exactly, so 97 is prime.

Does it matter which factor pair I start my tree with?

No. Every route ends with the same primes — that is the uniqueness of prime factorisation.

Can I find the HCF by listing factors instead?

Yes, for small numbers. Prime factors are faster and more reliable for larger numbers, and are required when the numbers are given in index form.

Key takeaways

Factors divide exactly; multiples are in the times table; primes have exactly two factors.
Use a factor tree to write a number as a product of primes in index form.
HCF: multiply the shared primes (lowest powers).
LCM: multiply all the primes (highest powers).
"Largest that fits into both" → HCF; "first time together" → LCM.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 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 33 marks

Write 180 as a product of prime factors. Give your answer in index form.[3]

Question 2Grade 43 marks

Burgers come in packs of 12. Buns come in packs of 18. Kai wants exactly the same number of burgers and buns.

What is the smallest number of packs of burgers he can buy?[3]

Question 3Grade 52 marks

$X = 2^{4} \times 3^{2} \times 7$ and $Y = 2^{2} \times 3^{3} \times 5$ . Find the HCF of $X$ and $Y$ , and the LCM of $X$ and $Y$ . Leave your answers in index form.[2]

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Factors, Multiples & Primes

Factors, multiples and primes

Prime factorisation

HCF and LCM

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