What is N4: Factors, multiples and primes?

A prime has exactly two factors, so 1 is not prime. Every integer above 1 is a product of primes in exactly one way, which is why the prime factorisation is a reliable route to the highest common factor and lowest common multiple.

How many marks are N4 questions worth?

In verified past-paper appearances, N4 questions on Edexcel GCSE Maths 1MA1 have been worth 1–4 marks.

Which past exam papers has N4 appeared in?

N4 has appeared in 45 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Prime numbers, factors (divisors), multiples, common factors and multiples, highest common factor, lowest common multiple, prime factorisation with product notation and unique factorisation theorem

Factors, multiples and primes

Worked answers, methods and verified real exam appearances for N4 on Edexcel GCSE Maths 1MA1.

Definition

Specification N4: use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple and prime factorisation, including using product notation and the unique factorisation theorem.

Explanation

A prime has exactly two factors, so $1$ is not prime. Every integer above $1$ is a product of primes in exactly one way, which is why the prime factorisation is a reliable route to the highest common factor and lowest common multiple.
Find it with a factor tree or repeated division by primes, then write the answer as a product using index notation: $500=2^2\times5^3$ . A list of primes without multiplication signs is not a product and drops the accuracy mark.
For two numbers, a Venn diagram of their prime factors puts the shared primes in the overlap: the HCF is the product of the overlap and the LCM is the product of everything in the diagram.
Given numbers already in index form, the HCF takes the lower power of each prime and the LCM the higher.
Read the question twice, because HCF and LCM are routinely swapped.

Worked example

180n

is a cube number, where

n

is a positive integer. Work out the smallest possible value of

n

(3)

(Total for Question 1 is 3 marks)

Working

1.Write $180$ as a product of prime factors: $180=2^2\times3^2\times5$ .
2.A cube number has every index a multiple of $3$ , so the product needs one more $2$ , one more $3$ and two more $5$ s.
3.The smallest such $n$ is $2\times3\times5^2=150$ .
4.Check: $180\times150=27\,000=30^3$ .

Answer: $n=150$

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Additional guidance
1	for a complete method to find the prime factors of $180$ , eg $2^2\times3^2\times5$ or $2,2,3,3,5$	M1	Condone the inclusion of 1 for this mark
	for identifying the factors needed, eg $2\times3\times5\times5$ or $2\times3\times5^2$	M1	oe
	for $150$	A1	cao

Common mistakes

Don't include $1$ in a list of prime numbers.
Don't stop a factor tree with a composite number at an endpoint.
Don't use the larger exponents for the HCF instead of for the LCM.

Mark-scheme wording to use

“for a complete method to find prime factors, could be shown on a complete factor tree with no more than one error or by division by prime factors with no more than one error” — Pearson 1MA1, November 2022 Paper 1H, Q1
“for listing at least 4 correct factors (with no more than 1 incorrect) of 63 or 105, could be in factor pairs or for the prime factors of 63 (3, 3, 7) or 105 (3, 5, 7)” — Pearson 1MA1, June 2024 Paper 3H, Q1
“M1 for a complete method to find prime factors; could be shown on a complete factor tree with no more than one error or by division by prime factors with no more than one error … A1 for 2 × 2 × 31 oe” — Pearson 1MA1, June 2022 Paper 1H, Q2

What the examiner wants

A product, and the right one of HCF or LCM. The June 2024 Paper 2H report says students who listed the prime factors rather than writing them as a product scored one mark, and that asked for an LCM from index form many gave the HCF of $6$ ; the Paper 3H report notes a significant proportion gave the LCM of $315$ when the HCF of $21$ was asked for.

Exam tip

For HCF or LCM questions, write both prime-power decompositions first so the method marks remain available.

Worked practice

Tier 1 · Easy

Write

756

as a product of its prime factors.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	$756=2^2\times3^3\times7$	2	Divide successively by primes: $756=2\times378=2^2\times189=2^2\times3^3\times7$ .

Tier 2 · Standard

Find both the HCF and the LCM of

84

and

126

(3)

(Total for Question 2 is 3 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	HCF $=42$ LCM $=252$	3	Use $84=2^2\times3\times7$ and $126=2\times3^2\times7$ . The smaller common powers give $2\times3\times7=42$ . The largest powers give $2^2\times3^2\times7=252$ .

Tier 3 · Hard

180n

is a cube number, where

n

is a positive integer. Find the smallest possible value of

n

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	$n=150$	4	Prime factorise $180=2^2\times3^2\times5$ . A cube needs every exponent to be a multiple of $3$ , so multiply by $2\times3\times5^2$ . Therefore $n=2\times3\times25=150$ , and $180n=27000=30^3$ .

Tier 1 · Easy

Write down all the prime factors of

42

(1)

(Total for Question 4 is 1 mark)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	$2$ , $3$ and $7$	1	$42=2\times3\times7$ , and each of $2$ , $3$ and $7$ is prime.

Tier 2 · Standard

Find the smallest multiple of both

12

and

18

that is greater than

100

(2)

(Total for Question 5 is 2 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$108$	2	$12=2^2\times3$ and $18=2\times3^2$ , so their LCM is $2^2\times3^2=36$ . The first multiple of $36$ greater than $100$ is $36\times3=108$ .

Tier 3 · Hard

n

is a positive multiple of

15

less than

100

. The HCF of

60

and

n

15

. Work out all the possible values of

n

(3)

(Total for Question 6 is 3 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	$n=15,45,75$	3	The multiples of $15$ below $100$ are $15,30,45,60,75,90$ . Their HCFs with $60$ are respectively $15,30,15,60,15,30$ , so the required values are $15$ , $45$ and $75$ .

Tier 2 · Standard

A club has

96

red counters,

144

blue counters and

168

green counters. The counters are put into the greatest possible number of identical bags with none left over. Work out the total number of counters in each bag.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	$17$ counters	3	The HCF of $96$ , $144$ and $168$ is $24$ , so there are $24$ bags. Each bag contains $96\div24=4$ red, $144\div24=6$ blue and $168\div24=7$ green counters. This is $4+6+7=17$ counters.

Tier 3 · Hard

n

is a positive integer. The HCF of

n

and

72

18

. The LCM of

n

and

72

360

. Work out

n

. Your method must be shown.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	$n=90$	4	Since the HCF is $18$ , $n$ is a multiple of $18$ . Since the LCM is $360$ , $n$ is a factor of $360$ . The multiples of $18$ that divide $360$ are $18,36,72,90,180,360$ . Checking these conditions leaves $n=90$ , because $\operatorname{HCF}(90,72)=18$ and $\operatorname{LCM}(90,72)=360$ .

Tier 3 · Hard

A whole number leaves a remainder of

5

when it is divided by

18

and also when it is divided by

24

. Work out the smallest three-digit number with this property.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	$149$	4	Subtracting $5$ from the number must give a common multiple of $18$ and $24$ . Since $18=2\times3^2$ and $24=2^3\times3$ , their LCM is $2^3\times3^2=72$ . The first multiple, $72$ , gives $77$ , which is not three-digit; the next gives $144+5=149$ .

Q10

Tier 3 · Hard

The number

2^7\times3^5\times5^2

has a factor that is a cube number. Work out its greatest factor that is a cube number.

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	$1728$	4	Every exponent in a cube number is a multiple of $3$ . The greatest allowed exponents are therefore $6$ for the factor $2$ , $3$ for the factor $3$ and $0$ for the factor $5$ . The required factor is $2^6\times3^3=64\times27=1728$ .

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2023-11	2F	Q1	1	Allowed	Foundation	QP MS
2024-06	3F	Q22	2	Allowed	Foundation	QP MS
2023-11	1H	Q3	4	Non-calculator	Higher	QP MS
2023-06	2F	Q5	1	Allowed	Foundation	QP MS
2019-11	1F	Q20	3	Non-calculator	Foundation	QP MS
2023-06	2F	Q21	2	Allowed	Foundation	QP MS
2021-11	2F	Q6	2	Allowed	Foundation	QP MS
2019-11	3F	Q1	1	Allowed	Foundation	QP MS
2023-11	3F	Q24	3	Allowed	Foundation	QP MS
2022-11	1H	Q8	2	Non-calculator	Higher	QP MS
2023-06	1F	Q7	2	Non-calculator	Foundation	QP MS
2019-06	2F	Q3	1	Allowed	Foundation	QP MS
2019-06	1H	Q3	2	Non-calculator	Higher	QP MS
2024-06	3H	Q1	2	Allowed	Higher	QP MS
2022-06	2F	Q4	1	Allowed	Foundation	QP MS
2021-11	3F	Q2	1	Allowed	Foundation	QP MS
2022-06	3F	Q21	2	Allowed	Foundation	QP MS
2024-11	1F	Q4	1	Non-calculator	Foundation	QP MS
2024-06	2F	Q4	1	Allowed	Foundation	QP MS
2022-11	3F	Q5	1	Allowed	Foundation	QP MS
2021-11	2F	Q22	4	Allowed	Foundation	QP MS
2023-06	3F	Q24	3	Allowed	Foundation	QP MS
2024-06	3F	Q12	2	Allowed	Foundation	QP MS
2022-06	1H	Q2	2	Non-calculator	Higher	QP MS
2023-11	3H	Q5	3	Allowed	Higher	QP MS
2019-06	1F	Q4	1	Non-calculator	Foundation	QP MS
2022-06	1F	Q24	2	Non-calculator	Foundation	QP MS
2019-11	3F	Q4	1	Allowed	Foundation	QP MS
2023-11	1F	Q20	4	Non-calculator	Foundation	QP MS
2023-06	2H	Q2	2	Allowed	Higher	QP MS
2022-11	1F	Q4	1	Non-calculator	Foundation	QP MS
2019-06	1F	Q24	2	Non-calculator	Foundation	QP MS
2024-11	3F	Q16	4	Allowed	Foundation	QP MS
2019-11	3F	Q20	4	Allowed	Foundation	QP MS
2022-11	1H	Q1	3	Non-calculator	Higher	QP MS
2024-11	3F	Q3	1	Allowed	Foundation	QP MS
2024-06	2F	Q21	3	Allowed	Foundation	QP MS
2024-06	2H	Q2	3	Allowed	Higher	QP MS
2022-06	3F	Q3	1	Allowed	Foundation	QP MS
2019-11	1H	Q1	3	Non-calculator	Higher	QP MS
2022-11	1F	Q19	3	Non-calculator	Foundation	QP MS
2019-06	3F	Q2	1	Allowed	Foundation	QP MS
2023-06	3H	Q5	3	Allowed	Higher	QP MS
2019-06	2H	Q17	4	Allowed	Higher	QP MS
2021-11	2H	Q2	4	Allowed	Higher	QP MS

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