What is N4: Factors, multiples and primes?

A prime number has exactly two positive factors, 1 and itself; 1 is not prime. Every integer greater than 1 has a unique prime factorisation apart from the order of its factors.

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.

Explanation

A prime number has exactly two positive factors, $1$ and itself; $1$ is not prime. Every integer greater than $1$ has a unique prime factorisation apart from the order of its factors.
Produce it with a factor tree or repeated division, then write repeated factors using powers. For two or more numbers, the highest common factor uses every shared prime with its smallest exponent.
The lowest common multiple uses every prime present with its largest exponent.
This method avoids double-counting factors and also supports problems about making a product into a square or cube.
Examiners require the factorisation to contain primes only.

Worked example

$180n$ is a cube number, where $n$ is a positive integer. Find the smallest possible $n$ .

1.Prime factorise: $180=2^2\times3^2\times5$ .
2.A cube needs exponents in multiples of $3$ , so supply $2^1\times3^1\times5^2$ .
3.Evaluate $n=2\times3\times25=150$ and check $180n=27000=30^3$ .

Answer: $n=150$ .

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.

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

Other points in N Number

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