1 Integers

1.4 Highest common factors

Worked example 1.4 Key words
a Find the factors of 24. common factor
conjecture
b Find the highest common factor (HCF) of 24 and 80.
consecutive
Answer factor
a 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6 highest common
factor (HCF)
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
b You must find the highest factor of 24 that is also a factor of 80.
8 is a factor of 80 because 80 ÷ 8 = 10.
12 is not a factor of 80 because 80 ÷ 12 = 6 remainder 4.
24 is not a factor of 80.
The highest common factor of 24 and 80 is 8.

Exercise 1.4
Focus
1 Find the factors of:
a 21 b 32 c 50 d 72 e 43
2 Find the factors of:
a 51 b 52 c 53 d 54 e 55
3 a Find the common factors of 16 and 28.
b Find the highest common factor of 16 and 28.
4 a Find the common factors of 30 and 45.
b Find the highest common factor of 30 and 45.

14
 1.4 Highest common factors

Practice
5 Find the highest common factor of:
a 18 and 21 b 18 and 27 c 18 and 36
6 Find the highest common factor of:
a 27 and 45 b 50 and 75 c 40 and 72 d 24 and 35
7 Find the highest common factor of:
a 70 and 77 b 70 and 85 c 70 and 84
8 a Find the highest common factor of 32 and 40.
b Use your answer to part a to simplify the fraction 32 .
40
9 a Find the highest common factor of 52 and 91.
b There are 91 rooms in a hotel. 52 rooms are reserved. What fraction of
the rooms are reserved?

Challenge
10 Two numbers have a highest common factor of 5. The two numbers add up to
35. Show that there are three possible pairs of values for the two numbers.
11 The HCF of two numbers is 4. Both numbers are more than 4 and less
than 30.
a Show that the numbers could be 8 and 12.
b Show that the numbers are not 8 and 16.
c Find all the other possible values of the two numbers.
12 The HCF of two numbers is 3. One of the numbers is 9. What are the possible
values of the other number?
13 a Find the highest common factor of:
i 9 and 10 ii 20 and 21 iii 32 and 33
b 9 and 10 are consecutive numbers. 20 and 21 are consecutive numbers.
Use part a to make a conjecture about the highest common factor of two
consecutive numbers.
c What is the lowest common multiple of two consecutive numbers?

15
1 Integers

1.5 Tests for divisibility

Worked example 1.5 Key words
Use tests for divisibility to show that 3948 is divisible by 3 and 6 but divisible
not by 9. tests for
divisibility
Answer

The sum of the digits is 3 + 9 + 4 + 8 = 24.

24 is divisible by 3, so 3948 is also divisible by 3.
3948 is even and divisible by 3, so 3948 is also divisible by 6.
24 is not divisible by 9, so 3948 is also not divisible by 9.

Exercise 1.5
Focus
1 Use tests for divisibility to show that 5328 is divisible by 4 and by 9.
2 a Show that 2739 is divisible by 11.
b When the digits are reversed, the number is 9372.
Is 9372 divisible by 11? Give a reason for your answer.
3 a Show that 67 108 is divisible by 4.
b Is 67 108 divisible by 8? Give a reason for your answer.
4 The number 3812* is divisible by 3. The final digit is missing. What can you say
about the missing digit?

Practice
5 What integers less than 12 are factors of 7777?
6 a  se the digits 5, 4, 2 and 1 to make a number that is divisible by:
U
i 5 ii 3
b Can you arrange the digits 5, 4, 2 and 1 to make a number that is divisible by:
i 9? ii 11?
7 322 is divisible by 7. Use this fact to find a number that is divisible by 7, 2 and 3.
8 Find the smallest positive integer that is not a factor of 2520. Give reasons for
your answer.

16
 1.6 Square roots and cube roots

9 Here are some numbers where all the digits are 9:

9  99  999  9999  99999  ...
In numbers where all the digits are 9, which are multiples of 11?

Challenge
10 A number is divisible by 15 if it is divisible by 3 and 5.
a Show that 7905 is divisible by 15.
b The number 208** is divisible by 15. Find the possible values
of the two missing digits.
11 Find three numbers less than 20 that are factors of 3729.
Give reasons for your answers.
12 Show that 8897 is divisible by only one number between 1 and 12.
13 The numbers 4, 5, 6, … are consecutive numbers; for example, 4567
is a number with four consecutive digits.
Find all the numbers with four consecutive digits that are divisible by:
a 2 b 3 c 5 d 11

1.6 Square roots and cube roots

Worked example 1.6 Key words
Work out 3 125 − 49 . consecutive
cube number
Answer
cube root
53 = 5 × 5 × 5 = 125, so 3 125 = 5. square number
72 = 7 × 7 = 49, so 49 = 7. square root
3
125 − 49 = 5 − 7 = −2

Exercise 1.6
Focus
1 Work out:
a 32 + 42 b 6² + 72 c 92 + 102

17
1 Integers

2 Work out:
a 64 b 100 c 225 d 169
3 Work out:
a 13 + 23 b 33 + 53 c 43 − 23
4 Work out:
a 64 − 3 64 b 25 − 3 125 c 3
27 − 16
5 Work out:
a 3
216 b 3
512 c 3
1000 d 3
1728

Practice
6 Copy and complete:
a = 20 b = 25 c = 30 d = 35
7 Copy and complete:
a 3 =6 b 3 = 10 c 3 = 11 d 3 = 15
8 Find the integer that is closest to:
a 38 b 220 c 3
70
9 45 is between 6 and 7.
Write down a similar statement for:
a 90 b 135
10 Mustafa thinks of a number. The number is between 100 and 200. The square
root of the number is a multiple of 3.
What is Mustafa’s number?
11 a  ind all the numbers between 100 and 200 that have an integer
F
square root.
b Find all the numbers between 100 and 200 that have an integer cube root.

Challenge
12 Find the highest common factor of 12 + 22 + 32 and 42 + 52 + 62.
13 Jiale thinks of a number. She works out the square root of the
number. Then she works out the cube root of the square root of the
number. The answer is 2.
a Find Jiale’s number.
b Show that Jiale gets the same answer if she finds the cube root first and then
the square root.

18
1 Integers
1.1 Factors, multiples and primes
Exercise 1.1 Key words

Focus factor tree

highest common
1 a Draw a factor tree for 250 that starts with 2 × 125. factor (HCF)
b Can you draw a different factor tree for 250 that starts with lowest common
2 × 125? Give a reason for your answer. multiple (LCM)
c Draw a factor tree for 250 that starts with 25 × 10. prime factor
d Write 250 as a product of its prime factors.
2 a Draw a factor tree for 300.
b Draw a different factor tree for 300.
c Write 300 as a product of prime numbers.
3 a Write as a product of prime numbers
i 6 ii 30 iii 210
b What is the next number in this sequence? Why?
4 Work out
a 2×3×7 b 22 × 32 × 72 c 23 × 33 × 73
5 a Draw a factor tree for 8712.
b Write 8712 as a product of prime numbers.
6 Write each of these numbers as a product of its prime factors.
a 96 b 97 c 98 d 99

Practice
7 Write as a product of prime numbers
a 70 b 702 c 703
8 a Write each square number as a product of its prime factors.
i 9 ii 36 iii 81
iv 144 v 225 vi 576
vii 625 viii 2401

7
1 Integers

b When a square number is written as a product of prime numbers,

what can you say about the factors?
c 176 400 = 24 × 32 × 52 × 72
Use this fact to show that 176 400 is a square number.
9 315 = 32 × 5 × 7 252 = 22 × 32 × 7 660 = 22 × 3 × 5 × 11
Use these facts to find the highest common factor of
a 315 and 252 b 315 and 660 c 252 and 660
10 60 = 22 × 3 × 5 72 = 23 × 32 75 = 3 × 52
Use these facts to find the lowest common multiple of
a 60 and 72 b 60 and 75 c 72 and 75
11 a Write 104 as a product of its prime factors.
b Write 130 as a product of its prime factors.
c Find the HCF of 104 and 130.
d Find the LCM of 104 and 130.
12 a Write 135 as a product of prime numbers.
b Write 180 as a product of prime numbers.
c Find the HCF of 135 and 180.
d Find the LCM of 135 and 180.

Challenge
13 a Write 343 as a product of prime numbers.
b Write 546 as a product of prime numbers.
c Find the HCF of 343 and 546. d Find the LCM of 343 and 546.
14 Find the LCM of 42 and 90.
15 a Find the HCF of 168 and 264.
b Find the LCM of 168 and 264.
16 a Show that the LCM of 48 and 25 is 1.
b Find the HCF of 48 and 25.
17 The HCF of two numbers is 6. The LCM of the two numbers is 72.
What are the two numbers?

8
 1.2 Multiplying and dividing integers

1.2 Multiplying and

dividing integers
Exercise 1.2 Key word

Focus integer

1 Copy this sequence of multiplications and add three more

multiplications in the sequence.
7 × −4 = −28   5 × −4 = −20   3 × −4 = −12   1 × −4 = −4
2 Work out
a −5 × 8 b −5 × −8
c −9 × −11 d −20 × −6
3 Put these multiplications into two groups.

A −12 × −3 B (−6)2 C −4 × 9

D 18 × 2 E 9 × −4 F −4 × −9
4 Copy and complete this multiplication table.

× −4 −9
−6
5 −45
−8 −16

5 Work out
a (3 + 4) × 5 b (3 + −4) × 5 c (−3 + −4) × −5 d (3 + −4) × −5

Practice
6 Estimate the answers by rounding numbers to the nearest integer.
a −2.9 × −8.15 b 10.8 × −6.1 c (−8.8)2 d (−4.09)2
7 Show that (−6)2 + (−8)2 − (−10)2 = 0
8 This is a multiplication pyramid.
Each number is the product of the two numbers below.
For example, 3 × −2 = −6
a Copy and complete the pyramid. –6
b Show that you can change the order of the numbers on the 3 –2 –1 4
bottom row to make the top number 3456.

9
1 Integers

9 a The product of two integers is −6.

Find all the possible values of the two integers.
b The product of two integers is 6.
Find all the possible values of the two integers.
10 a Here is a multiplication: −9 × −7 = 63
Write it as a division in two different ways.
b Here is a different multiplication: 12 × −7 = −84
Write it as a division in two different ways.
11 Work out
a 42 ÷ −7 b −50 ÷ −10 c 27 ÷ −3 d −52 ÷ −4 e 60 ÷ −5
12 Estimate the answers by rounding numbers to the nearest 10.
a 92 ÷ −28.5 b −41 ÷ −18.9 c 83.8 ÷ −11.6 d −77 ÷ 19

Challenge
13 Copy and complete this multiplication pyramid.
270

15

–3

–3

14 Find the value of y.

a −8 × y = 48 b y × −3 = −36
c −10 × y = 120 d y × −5 = −40
15 Find the value of z.
a z ÷ −4 = −8 b z ÷ −2 = 20
c −36 ÷ z = 9 d 30 ÷ z = −6
16 a Here is a statement: −3 × (−6 × −4) = (−3 × −6) × −4
Is it true or false? Give a reason to support your answer.
b Here is a statement: −24 ÷ (−4 ÷ −2) = (−24 ÷ −4) ÷ −2
Is it true or false? Give a reason to support your answer.

10
 1.3 Square roots and cube roots

1.3 Square roots and cube roots

Exercise 1.3 Key words

Focus natural numbers

rational numbers
1 Work out
a 142 b (−14)2 c (−20)2 d (−30)2
2 Work out
a 43 b (−6)3 c (−10)3 d (−1)2 + (−1)3
3 If possible, work out
a −64 b 3
−64 c 3
−125 d 3
−729

Practice
4 Solve each equation.
a x2 = 25 b x2 = 225 c x2 − 81 = 0 d x2 + 121 = 0
5 Solve each equation.
a x3 = 216 b x3 = −216 c x3 + 1000 = 0 d x3 + 8000 = 0
6 232 = 529 and 233 = 12 167
Use these facts to solve the following equations.
a x2 = 529 b x2 + 529 = 0 c x3 = 12 167 d x3 + 12 167 = 0
7 Write whether each statement is true or false.
a 9 is a rational number
b −9 is a natural number
c 99 is an integer
d −999 is both an integer and a rational number
e 9999 is both a natural number and a rational number

Challenge
8 a Copy and complete this table.

x −3 −2 −1 0 1 2
x +x
2
0
x3 + x 0

b Use the table to solve these equations.

i x2 + x = 2 ii x3 + x = 2

11
1 Integers

9 Here is an equation: x3 − x = 120

a Is x = 5 a solution? Give a reason for your answer.
b Is x = −5 a solution? Give a reason for your answer.
10 a Write 64 as a product of its prime factors.
b Show that 64 is a square number and a cube number.
c Write 729 as a product of prime numbers.
d Show that 729 is both a square number and a cube number.
e Find another integer that is both a square number and a cube number.
11 Look at the following solution of the equation x6 = 64

x6 = 64
so x3 = 8
so x=2

There is an error in this solution. Write a corrected version.

1.4 Indices
Key words
Exercise 1.4 index
power
Focus
1 Write as a single power
a 32 × 3 b 7 × 73 c 12 × 125 d 154 × 15
2 Write as a single power
a 63 × 63 b 105 × 102 c 36 × 33 d 143 × 144
3 a Show that 20 + 21 + 22 + 23 = 24 – 1
b Can you find a similar expression for 20 + 21 + 22 + 23 + 24 + 25?
c Read what Zara says:

30 + 31 + 32 + 33 = 34 − 1

Is she correct? Give a reason for your answer.

12
 1.4 Indices

Practice
4 Write as a single power
a (5 )3 2
b (15 )
3 2
c (7 )3 3
d (3 ) 4 5

5 a Write 4 as a power of 2. b Write 43 as a power of 2.

c Write 93 as a power of 3.
6 54 = 625
Write as a power of 5
a 6252 b 6253 c 6254
7 Find the missing power.

a 42 × 4 = 45 b 74 × 7 = 76
c 153 × 15 = 156 d 15 × 154 = 154
8 Work out and write the answer in index form.
a 83 ÷ 8 b 56 ÷ 52 c 210 ÷ 22
d 36 ÷ 33 e 124 ÷ 124
9 Find the missing power of 6.
a 65 ÷ 6 = 62 b 68 ÷ 6 = 64
c 6 ÷ 62 = 66 d 6 ÷ 63 = 63

Challenge
10 Work out and write the answer in index form.
a 45 ÷ 23 b 94 ÷ 35 c 322 ÷ 26 d 272 ÷ 36
11 Write as a power of 5
a 125 b 1252 c 1254
12 12 = 20 736
4

Write as a power of 12
a 20 7362 b 20 7363 c 20 736
13 Read what Marcus says:

24 = 42 and so 34 = 43

Is Marcus correct? Give a

reason to support your answer.

13