Preview of Cambridge IGCSE International Mathematics Coursebook

Dear Cambridge Teacher,

Welcome to the first edition of our Cambridge IGCSE™ International Mathematics series, which supports the Cambridge IGCSE™ International Mathematics syllabus (0607) for examination from 2025. We have developed our first edition through extensive research with teachers around the world, to provide you and your learners with the support you need, where you need it. You can be confident that this series supports all aspects of the syllabus.

This Executive Preview contains sample content from the series, including:

• The table of contents from the coursebook

• A guide explaining how to use this series

• A guide explaining how to use this book

• Chapter 1 of the coursebook

We have ‘Worked examples’ in our coursebook as well as engaging exercise sets, to include a step-bystep process of working through a question or problem. A ‘Mathematics in Context’ feature demonstrates where students will use mathematics in real life.

Our ‘Mathematical connections’ feature connects mathematics topics within the coursebook to enable learners to build upon their knowledge as they progress through the course. A non-calculator icon is flagged next to some questions to encourage students to work through problems mentally, helping prepare students for non-calculator questions in their assessments . We have also included investigations and modelling in our coursebook to help improve learners’ problem -solving skills. Further, we have signalled which is Core and which is Extended content so that questions are pitched at the appropriate level for students.

The digital teacher's resource provides interactive ideas for lessons and focuses on key syllabus points and pedagogical approaches. There is also advice on supporting language in mathematics and common student misconceptions.

Finally, we are happy to introduce Cambridge Online Mathematics, hosted on our Cambridge GO platform. Cambridge Online Mathematics provides enhanced teacher and student support; it can be used to create virtual classrooms enabling you to blend print and digital resources into your teaching, in the classroom or as homework. The platform is easy to use, tablet-friendly and flexible.

We hope you enjoy this new series of resources. Visit our website to view the full series or speak to your local sales representative. You can find their details here: c cambridge org/gb /education/find -your -sales-consultant

With best wishes from the Cambridge team,

Arifah A. Khan Commissioning Editor for Cambridge IGCSE ™ International Mathematics Cambridge University Press

How to use this series

This suite of resources supports students and teachers following the Cambridge IGCSE™ International Mathematics syllabus (0607) for examination from 2025. The syllabus content has been addressed using a range of up-to-date metacognition techniques ensuring that mathematics is taught to students to develop a holistic understanding of mathematics. The components in the series are designed to work together.

The coursebook is designed for students to use in class with guidance from the teacher. There are 24 chapters including a modelling chapter that has been included to accommodate the updated syllabus. There are also a series of investigations throughout the chapters to develop problem-solving skills and to meet syllabus requirements. A variety of metacognition techniques have been used such as discussion activities and re ection features to encourage students to deepen their mathematical approach. Mathematics in Context has been introduced to help students understand the real use cases of mathematics in everyday life. The Mathematical Connections feature links topics in the coursebook to create holistic view of mathematics for learners. Access to Cambridge Online Mathematics is provided to access the coursebook digitally. A teacher account can be set up and online classes can be created.

The digital Teacher's Resource is succinct and offers lesson ideas such as starters, main activities and plenaries for teachers. How to identify and address common misconceptions is provided as well as language support to ensure that teachers and learners are well-supported in the classroom. Differentiation ideas are also provided from helping learners who need more support to challenging students. The Teacher's Resource is in-line with the coursebook and covers the same topics. Answers for the Teacher's Resource and coursebook are provided on the Cambridge GO platform.

Cambridge IGCSE™ International Mathematics

Digital Teacher’s Resource

Cambridge IGCSE™ International Mathematics

Peter Blythe, Emma Low, Andrew Manning, Karen Morrison, Raju Taniparti & Jasmine S. M. Teo

Cambridge Online Mathematics

Discover our enhanced digital mathematics support for Cambridge Lower Secondary, Cambridge IGCSE™ and Cambridge International AS & A Level Mathematics – endorsed by Cambridge Assessment International Education.

Available in 2023

New content to support the following syllabuses:

• Cambridge IGCSE Mathematics

• Cambridge IGCSE International Mathematics

• Cambridge IGCSE and O Level Additional Mathematics

Features can include:

• Guided walkthroughs of key mathematical concepts for students

• Teacher-set tests and tasks with auto–marking functionality

• A reporting dashboard to help you track student progress quickly and easily

• A test generator to help students practise and refine their skills – ideal for revision and consolidating knowledge

Free trials

A free trial will be available for Cambridge IGCSE Mathematics in 2023. In the meantime, please visit https://bit.ly/3TUGl4l for a free trial of our Cambridge Lower Secondary and Cambridge International AS & A Level Mathematics versions.

How to use this book

Throughout this book, you will notice lots of different features that will help your learning. These are explained below.

IN THIS CHAPTER YOU WILL

These set the scene for each chapter, help with navigation through the coursebook and indicate the important concepts in each topic.

Extended content

Where content is intended for students who are studying the Extended content of the syllabus as well as the Core, this is indicated using the arrow and the bar, as on the left here.

GETTING STARTED

This contains questions and activities on subject knowledge you will need before starting this chapter.

KEY WORDS

The key vocabulary appears in a box at the start of each chapter, and is highlighted in the text when it is first introduced. You will also find definitions of these words in the Glossary at the back of this book.

TIP

The information in this feature will help you complete the exercises, and give you support in areas that you might find difficult.

Exercises

Appearing throughout the text, exercises give you a chance to check that you have understood the topic you have just read about and practise the mathematical skills you have learned. You can find the answers to these questions in the back of the Coursebook.

INVESTIGATION/DISCUSSION

These boxes contain questions and activities that will allow you to extend your learning by investigating a problem, or by discussing it with classmates.

WORKED EXAMPLES

These boxes show you the step-by-step process to work through an example question or problem, giving you the skills to work through questions yourself.

MATHEMATICS IN CONTEXT

This feature presents real-world examples and applications of the content in a chapter, encouraging you to look further into topics.

REFLECTION

These activities ask you to think about the approach that you take to your work, and how you might improve this in the future.

This icon shows you where you should complete an exercise without using your calculator.

Past paper questions

Past paper questions allow you to practise your exam skills. Answers to these questions can be found in the back of the Coursebook.

MATHEMATICAL CONNECTIONS

This feature will help you to link content in the chapter to what you have already learned, and highlights where you will use your understanding again in the course.

SUMMARY

There is a summary of key points at the end of each chapter.

SELF/PEER ASSESSMENT

At the end of some exercises you will find opportunities to help you assess your own work, or that of your classmates, and consider how you can improve the way you learn.

Chapter 1 Number

IN THIS CHAPTER YOU WILL:

• convert between numbers and words

• identify natural numbers, integers, prime numbers, square numbers, cube numbers, triangle numbers, prime factors and rational and irrational numbers

• find the reciprocal of a number

• find the lowest common multiple (LCM) and highest common factor (HCF) of two numbers

SAMPLE

• calculate with squares, square roots, cubes, and cube roots of numbers and with other powers and roots of numbers

• round values to specific numbers of decimal places or significant figures

• estimate calculations by rounding numbers to significant figures.

GETTING STARTED

1 Write down the whole numbers:

a between 10 and 20

b greater than 25 but less than or equal to 30.

2 In the list of numbers: 25, 1 2 , 0.125

Write down:

a decimal b fraction c whole number.

3 List the factors of:

a 12 b 30 c 42

4 Write down the first five multiples of:

a 3 b 5 c 11

1.1 Types of number

Being able to identify different types of number will help you to understand how number operations work and to evaluate problems involving numbers.

Natural numbers

The natural numbers are: 0, 1, 2, 3, 4, 5, 6, Natural numbers have no decimal or fractional parts.

Converting between numbers and words

Natural numbers have different place values. These place values allow you to convert natural numbers into words.

This table summarizes the place values for the number 1 234 567 890.

KEY WORDS

natural numbers

prime number

composite number

root integer

square number

cube number

irrational number

reciprocal triangle number

lowest common multiple (LCM)

highest common factor (HCF)

TIP

Some books define the natural numbers starting from 1, but in this course the natural numbers start from zero.

You write 1 234 567 890 in words as one billion, two hundred and thirty-four million, five hundred and sixty-seven thousand, eight hundred and ninety.

WORKED EXAMPLE 1

1 Write the following numbers in words.

a 20 316

b 7 000 000 009

2 Write the following in numbers.

a One hundred and twenty-four thousand

b Fifteen million and seventy-eight

Answers

1 a First look at the place value of each digit in the number 20 316.

Digit Place value

20 316 six ones

20 316 one ten

20 316 three hundreds

20 316 zero thousands

20 316 twenty thousands

Starting from the highest place value, you have twenty thousand, three hundred and sixteen.

b In the number 7 000 000 009, only the digits 7 and 9 are non-z ero.

Digit Place value

7 000 000 009 nine ones

7 000 000 009 seven billions

Starting from the highest place value, you have seven billion and nine.

2 a Consider the place value of the numbers in words.

Words NumbersRemarks

One hundred and twenty-four thousand. 100 000 A thousand is 3 zeroes, so a hundred thousand is 100 with 3 zeroes.

One hundred and twenty-four thousand. 24 000

Twenty-four thousand is 24 with 3 zeroes.

Add up the numbers: 100 000 + 24 000 = 124 000

So, one hundred and twenty-four thousand in numbers is 124 000.

TIP

When expressing a number in words, omit any place value with a digit zero.

CONTINUED

b Consider the place value of the numbers in words.

Words

NumbersRemarks

Fifteen million and seventy-eight. 15 000 000 A million has 6 zeroes, so fifteen million is 15 with 6 zeroes.

Fifteen million and seventy-eight 78

Add up the numbers: 15 000 000 + 78 = 15 000 078

So fifteen million and seventy-eight in numbers is 15 000 078.

Categorizing natural numbers

There are many different ways to categorize natural numbers. For example, natural numbers can be even or odd.

Even numbers are divisible by two.

The even numbers are: 0, 2, 4, 6, 8, 10, …

In general, the last digit of an even number is a multiple of 2. Numbers that are not even are odd numbers.

The odd numbers are: 1, 3, 5, 7, 9, 11,

Odd numbers are not divisible by two.

In general, the last digit of an odd number is 1, 3, 5, 7, or 9.

DISCUSSION 1

1 Work with a partner to determine if the following statements are true or false.

a The sum of two even numbers is always even.

b The sum of two odd numbers is always odd.

c The sum of an even and an odd number is always odd.

d The product of two even numbers is always even.

e The product of two odd numbers is always odd.

f The product of an even and an odd number is always even. You can also classify natural numbers using their factors.

2 Copy and complete this table. Write down all the factors of the given numbers. The factors of 12 have been written for you.

Copy and complete this table to group the given numbers into the following:

GroupNumber of different factorsNumbers

INumber(s) with exactly one factor

IINumber(s) with exactly two different factors

IIINumber(s) with more than two different factors

The numbers in group II are known as prime numbers

The numbers in group III are known as composite numbers

The numbers zero and one are neither prime nor composite.

REFLECTION

1 Why are zero and one neither prime nor composite?

2 Do you agree with the statement ‘If a natural number is not prime, then it must be composite’?

Prime numbers

A prime number is a natural number that has exactly two different factors, one and itself. A composite number is a natural number that has more than two different factors.

INVESTIGATION 1

Sieve

The Sieve of Eratosthenes is a method to find prime numbers. In this investigation, you will use the Sieve of Eratosthenes to find all the prime numbers from 2 to 100.

Start from 2 because 2 is the smallest prime number. Follow the given instructions.

Step 1: circle 2, the smallest prime number, and cross out all the multiples of 2 less than or equal to 100.

Step 2: circle the next smallest number that is not crossed out, in this case 3. Cross out all the multiples of 3 less than or equal to 100.

Step 3: circle the next smallest number that is not crossed out, in this case 5. Cross out all the multiples of 5 less than or equal to 100.

Step 4: circle the next smallest number that is not crossed out and cross out all multiples of that number less than or equal to 100.

Step 5: repeat the same process until all the numbers are either crossed out or circled.

Answer the following questions.

1 Write down all the numbers you circled.

2 These numbers are the prime numbers less than 100.

3 How many prime numbers less than 100 are there?

4 How many even prime numbers less than 100 are there?

5 How many odd prime numbers less than 100 are there?

You can use the same process to find prime numbers less than any number.

A number is prime if it is not divisible by any of the prime numbers less than or equal to the square root of the number.

It follows that a number is composite if it is divisible by any prime factor. You can use this result to determine whether a given number is a prime number.

WORKED EXAMPLE 2

Determine whether the following numbers are prime numbers.

a 291 b 269

Answers

a Step 1: use a calculator to find the square root of 291: √291 = 17.058 72… Step 2: identify all the prime numbers less than or equal to the square root of 291.

The prime numbers less than or equal to 17.058 72… are: 2, 3, 5, 7, 11, 13, 17.

Step 3: try dividing 291 by these prime numbers. If 291 is not divisible by any of the prime numbers identified in step 2, then it is a prime number.

Consider 2: Since 291 is not even, it is not divisible by 2.

Try 3: 291 ÷ 3 = 97, so 291 is divisible by 3.

Since 291 is divisible by 3, it is not a prime number.

b √269 = 16.401 21…

The prime numbers less than or equal to √269 = 16.401 21… are: 2, 3, 5, 7, 11, 13.

Consider 2: 269 is odd, so it is not divisible by 2.

Try 3: 269 ÷ 3 = 89.666 6…, so 269 is not divisible by 3.

Only even numbers are divisible by 2.

A number is divisible by 3 if the sum of the digits of the number is divisible by 3. The digits of 269 are 2, 6 and 9. 2 + 6 + 9 = 17 which is not divisible by 3, so 269 is not divisible by 3.

Try 5: The last digit of 269 is 9, which is neither 0 nor 5, so 269 is not divisible by 5.

Try 7: 269 ÷ 7 = 38.428 57…, so 269 is not divisible by 7.

Only numbers with a last digit of 0 or 5 are divisible by 5.

A number is divisible by 7 if twice the last digit of the number subtracted from the remaining number is divisible by 7.

The last digit of 269 is 9. 9 × 2 = 18. The remaining number is 26. 26 − 18 = 8. Since 8 is not divisible by 7, 269 is also not divisible by 7.

CONTINUED

Try 11: 269 ÷ 11 = 24.454 5…, so 269 is not divisible by 11.

A number is divisible by 11 if the alternating sum of the digits (alternate sum and difference) is divisible by 11. For 269, the alternating sum is 2 − 6 + 9 = 5. Since 5 is not divisible by 11, 269 is not divisible by 11.

Try 13: 269 ÷ 13 = 20.692 3…, so 269 is not divisible by 13.

A number is divisible by 13 if the difference between four times the last digit of the number and the remaining number is divisible by 13.

The last digit of 269 is 9. 9 × 4 = 36. The remaining number is 26. 36 − 26 = 10. Since 10 is not divisible by 13, 269 is also not divisible by 13.

Since 269 is not divisible by 2, 3, 5, 7, 11, or 13, which are all the primes less than or equal to √269 = 16.401 21…, then 269 must be a prime number.

WORKED EXAMPLE 3

If a and b are natural numbers such that a × b = 19, find the value of a + b

Answer

Since 19 is a prime number, the only factors of 19 are 1 and 19. So a and b must be 1 and 19 in whichever order. Then a + b = 1 + 19 = 20

Integers (positive, negative and zero)

The natural numbers are part of the integers. Integers are numbers that do not have fractional or decimal parts. Integers can be positive or negative or zero.

The positive integers are: 1, 2, 3, 4, 5, …, 100, …

The negative integers are: −1, −2, −3, …, −45, … 0 is an integer but it is neither positive nor negative.

Exercise 1.1

1 Write the following numbers in words.

a 540 018 b 9 000 342 c 41 020 679

d 3 000 000 853 e 9 000 231 038 f 60 582

g 6 500 453 684

MATHEMATICAL CONNECTIONS

You will learn about the four operations for calculations with integers in Chapter 2.

Write the following in numbers.

a Two million, six hundred and eighteen thousand, four hundred and twenty-two.

b Five billion, four hundred and sixty-one.

c Seven hundred and four thousand and thirty-seven.

d Eighteen million, one hundred and fifty-three thousand and six.

3 Write down all the prime numbers that are less than 100.

4 Determine if each of these numbers is a prime number.

a 173 b 129 c 237 d 281 e 383

5 If a and b are natural numbers such that a × b = 23, find the value of a + b

6 If a and b are natural numbers such that a × b = 89, find the value of a + b.

1.2 Other types of number

Index notation, square numbers and cube numbers

Index notation is a way of writing numbers when you multiply a number by itself one or more times.

For example,

you can write 3 × 3 as 3 2

You read 3 2 as ‘three to the power of two’, where the number three is the base and the number two is the power or exponent.

The base represents the number that you multiplied by itself. The power is the number of times you multiplied the base by itself. So:

2 × 2 × 2 × 2 × 2 = 2 5 reads as ‘two to the power of five’

3 × 3 × 3 × 3 = 3 4 reads as ‘three to the power of four’

5 × 5 × 5 = 5 3 reads as ‘five to the power of three’.

When the power is two, you can also say that the base is ‘squared’.

For example, you read 3 2 as ‘three squared’.

Numbers with the power of two are called square numbers.

Examples of square numbers are:

2 2 = 2 × 2 = 4 3 2 = 3 × 3 = 9 4 2 = 4 × 4 = 165 2 = 5 × 5 = 25

When the power is three, you can also say that the base is ‘cubed’.

For example, you read 5 3 as ‘five cubed’.

Numbers with the power of three are called cube numbers

Examples of cube numbers are:

2 3 = 2 × 2 × 2 = 8 3 3 = 3 × 3 × 3 = 27

MATHEMATICAL CONNECTIONS

Indices are discussed further in Chapter 6.

WORKED EXAMPLE 4

Write the following in index notation.

a 2 × 2 × 2 × 2 × 2 × 2 × 2 b a × a × a × a × a

Answers

a 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 7 The number two is multiplied by itself, so the base is two. The base two is multiplied by itself seven times, so the power is seven.

b a × a × a × a × a = a 5

WORKED EXAMPLE 5

Evaluate.

The number a is multiplied by itself, so the base is a. The base a is multiplied by itself five times, so the power is five.

a 172 b 63 c 114

Answers

a 17 2 = 17 × 17 = 289

You can use a calculator to evaluate the product of such a big number. To evaluate a square number, key in:

b 6 3 = 6 × 6 × 6 = 216

To use your calculator to evaluate 63, key in:

c 11 4 = 11 × 11 × 11 × 11 = 14 641

To use your calculator to evaluate 114, key in:

Rational numbers and irrational numbers

A rational number is a number that can be expressed in the form

where a and b are natural numbers and

0

Examples of rational numbers are:

All natural numbers and fractions are rational. Some decimals are rational. An irrational number is a number that cannot be expressed in the form a b where a and b are natural numbers and b ≠ 0.

Examples of irrational numbers are: √ 2 , √ 5 , 3 √ 7 , π In general, square roots and cube roots that are not exact are irrational.

WORKED EXAMPLE 6

0.13, 3 7 , √3 , π, 8, 13, 16, 41, 64

From this list of numbers, write down:

a the square numbers

b the prime numbers

c the cube numbers

d the irrational numbers.

Answers

You need to categorize the numbers.

You can express 0.13 in fractional form. 0.13 = 13 100 which is rational.

3

7 is itself a rational number.

√ 3 does not give you an exact value, so it is an irrational number.

The value of π is 3.14159265 It is not exact and cannot be expressed as a fraction. Therefore, it is irrational.

8 = 2 × 2 × 2 = 2 3, hence it is a cube number.

13 only has two distinct factors, 1 and 13, hence it is a prime number. 16 = 4 × 4 = 4 2, so it is a square number.

41 only has two distinct factors, 1 and 41, hence it is a prime number.

64 is a rather special number. 64 = 8 × 8 = 8 2 which makes it a square number, but 64 = 4 × 4 × 4 = 4 3, so 64 is also a cube number.

a The square numbers are 16 and 64.

b The prime numbers are 13 and 41.

c The cube numbers are 8 and 64.

d The irrational numbers are √3 and π.

MATHEMATICAL CONNECTIONS

You will look closer at square roots and cube roots in Section 1.5.

Reciprocals

A reciprocal is a multiplicative inverse such that the product of a number and its reciprocal will produce 1. To put it simply, the reciprocal of a number a is 1 a

For example:

The reciprocal of 8 is 1 8 (8 × 1 8 = 1)

The reciprocal of 1 3 is 3. ( 1 3 × 3 = 1)

The reciprocal of 3 7 is 7 3 ( 3 7 × 7 3 = 1)

Triangle numbers

Investigation 2 explores the triangle numbers.

INVESTIGATION 2

Triangle numbers

Look at this sequence of patterns of dots.

MATHEMATICAL CONNECTIONS

You will study sequences in Chapter 22.

1 Copy and complete the following sentence: The dots in each pattern form the shape of a ………. .

2 Draw Pattern 5 and Pattern 6.

3 Copy and complete this table.

4 Without drawing the diagram for Pattern 10, write down the:

a number of dots along one side of the shape

b total number of dots in the shape

c observation about the total number of dots.

CONTINUED

5 What is the relationship between the total number of dots in the shape and the last column, n(n + 1)?

6 Which pattern will have a total number of 496 dots? Explain how you get your answer.

The total number of dots are known as triangular numbers.

Triangle numbers are numbers that can make equilateral triangle patterns. Examples of triangle numbers are: 1, 3, 6, 10, etc.

SELF ASSESSMENT

List all the:

a square numbers

b cube numbers

c triangle numbers between 0 and 100.

Exercise 1.2

1 Express the following in index notation. a

d 2 × 5 × 3 × 2 × 3 × 3 × 2 × 5 × 13 × 13

2 0.35, 1 2 , √ 5 , π 3 , 27, 36, 43, 51, 81

From this list of numbers, write down:

a the square numbers

c the cube number

e the triangular number.

3 −5, 22 7 , 0, √9 , √11 , 17, 25, 37, 125

From this list of numbers, write down:

a the negative integer

c the prime numbers

e the irrational number.

4 Write down the reciprocal of each number.

b the prime number

d the irrational numbers

TIP

For Questions 2 and 3, you might not use all of the numbers in the list.

b the square number

d the cube number

a 5 b 20 c 1 4 d 2 5

1.3 Products of prime factors

You can express each integer greater than one as a product of prime factors. Worked example 7 demonstrates two methods you can use.

WORKED EXAMPLE 7

Express 60 as a product of prime factors. Answers Method 1: Factor tree

Step 1: first, split 60 into the product of two factors. You can use any two factors.

Step 2: check if each of the two factors are prime. If both factors are prime, you can stop.

If both factors are not prime, split each nonprime factor further into the product of two factors.

Repeat Step 2 until all the factors are prime.

The number is now expressed as a product of prime factors.

If any of the prime numbers appear more than once in the product, use index notation. In this example 2 × 2 = 22

Step 1: divide 60 by its smallest prime factor In this case two.

Step 2: write the result of the division in the right column. Continue to divide the number in the right column by the same factor if possible. Write the result on the right column.

Step 3: repeat Step 2. If the number in the right column is not divisible by the same factor, divide this number by the next biggest prime factor.

Repeat Steps 2 and 3 until the number is reduced to one.

Multiply the numbers in the left column to express the original number as a product of prime factors.

Again, use index notation for any repeated prime factors.

Some books call this procedure ‘prime factorisation’.

DISCUSSION 2

In groups of 3 or 4, research and discuss the differences between these types of prime number:

1 Twin primes and cousin primes

2 Fermat primes

3 Sophie Germain primes

4 Palindromes and palindromic primes

5 Coprime

6 Importance and application of prime numbers in real life.

Exercise 1.3

1 Express each number as a product of prime factors.

6 The whole number p is such that p × ( p + 22) is a prime number. Find:

p b the prime number.

7 The whole number p is such that p × ( p + 106) is a prime number. Find:

p b the prime number.

INVESTIGATION 3

The total number of distinct positive factors of any natural number.

1 a What are factors?

b List all the factors of 24.

c How many distinct positive factors does 24 have in total?

2 Alternative method to finding the number of distinct positive factors of a natural number, X

Step 1: express X as a product of prime factors.

X = p a × q b × r c

Step 2: add one to all the powers.

a + 1, b + 1, c + 1

Step 3: the product of the results in Step 2 will give the total number of distinct positive factors of X

Total number of distinct positive factors of X = (a + 1)(b + 1)(c + 1)

Verify that this alternative method works to find the total number of distinct positive factors of 24.

3 Consider the number 60. Use both methods from Questions 1 and 2 to verify the total number of distinct positive factors of 60.

4 Consider three different natural numbers.

Use both methods from Questions 1 and 2 to find the total number of distinct positive factors of these numbers.

Of your three numbers:

a one number must be prime

b one number must contain three prime factors when expressed as a product of prime factors

c the other number must have at least one power in the product of prime factors that is greater than one.

5 Explain why the alternative method works using the number 24.

6 Find the total number of distinct positive factors of 229 320.

1.4 Highest common factors (HCF) and lowest common multiples (LCM) Highest common factors (HCF)

You can use a listing method to find the highest common factor (HCF) of small numbers.

For example:

The factors of 12 are: 1, 2, 3, 4, 6 and 12.

The factors of 40 are: 1, 2, 4, 5, 8, 10, 20 and 40.

The common factors of 12 and 40 are 1, 2, and 4.

4 is the HCF of 12 and 40.

Worked example 8 demonstrates two more efficient methods to find the HCF of two numbers.

WORKED EXAMPLE 8

Find the HCF of 84 and 378.

Answers

Method 1: Repeated division by common prime factors

284, 378 42, 189

284, 378 342, 189 14, 63

284, 378

342, 189

714, 63 2, 9

Since 84 and 378 are even, they are both divisible by 2. Divide both numbers by 2.

42 is even but 189 is odd, so they are not both divisible by 2. Try 3.

14 is not divisible by 3, so try the next prime number. Both 14 and 63 are not divisible by 5, so try 7.

Both 14 and 63 are divisible by 7, so divide both numbers by 7.

2 is a prime number so it can only be divided by itself. You already used 2 as the common factor at the beginning of this method, so there is no common factor of 2 and 9.

The division stops here.

284, 378 342, 189

714, 63

2, 9

The product of the prime factors on the left gives the HCF of 84 and 378.

The HCF of 84 and 378 is 2 × 3 × 7 = 42

Method 2: Using products of prime factors

84 = 2 2 × 3 × 7

378 = 2 × 3 3 × 7

84 = 2 2 × 3 × 7

378 = 2 × 3 3 × 7

84 = 2 2 × 3 × 7

378 = 2 × 3 3 × 7 2 × 3 × 7

By using any one of the methods from Worked example 7, express both numbers individually as a product of prime factors.

Identify the common bases.

The common bases are 2, 3 and 7.

For the common bases, choose the smallest power of each base.

84 = 2 × 2 × 3 × 7

378 = 2 × 3 × 3 × 3 × 7

The product of these common factors gives the HCF.

The HCF of 84 and 378 = 2 × 3 × 7 = 42

TIP

You choose the smallest power for each base because the power indicates the number of times the base appears. To find the common prime factors, you need to consider the number of times the base is multiplied.

Lowest common multiples (LCM)

You can use a listing method to find the lowest common multiple (LCM) of two small numbers.

For example:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, Multiples of 5: 5, 10, 15, 20, 25, 30, …

The common multiples of 3 and 5 are 15, 30, 15 is the LCM of 3 and 5.

Worked example 9 demonstrates two more efficient methods to find the LCM of two numbers.

WORKED EXAMPLE 9

Find the LCM of 12 and 56.

Answers

Method 1: Repeated division

212, 56 6, 28

Since 12 and 56 are both even, you can divide both numbers by 2.

Both 6 and 28 are even. Continue to divide both numbers by 2. 212, 56

212, 56

3 is a prime number. There are no longer any common prime factors of 3 and 14. Instead of stopping here, continue to divide the numbers by prime factors until they are reduced to 1. Deal with one number at a time.

Since 14 is even, you can continue to divide 14 by 2. Leave 3 alone when you divide 14. 212, 56

Both 3 and 7 are prime numbers. Divide 3 and 7 individually by 3 and 7 to reduce them to 1.

When both numbers are reduced to 1, the product of the prime factors on the left will give you the lowest common multiple of 12 and 56.

2 3, 14 3 3, 7 7 1, 7 1, 1

The LCM of 12 and 56 is 2 3 × 3 × 7 = 168

Method 2: Using products of prime factors

12 = 2 2 × 3

56 = 2 3 × 7

12 = 2 2 × 3

56 = 2 3 × 7

12 = 2 2 × 3

56 = 2 3 × 7

2 3 × 3 × 7

By using any one of the methods from Worked example 7, express both numbers individually as a product of prime factors.

Identify all the bases, common or not. The bases are 2, 3 and 7.

If there are common bases, choose the biggest power of the common base.

12 = 2 × 2 × 3

56 = 2 × 2 × 2 × 7

The product of these factors gives the LCM.

The LCM of 12 and 56 = 2 3 × 3 × 7 = 168

Exercise 1.4

3 Find the HCF and LCM of the following pairs of numbers. Leave your answers as products of prime factors in index notation.

TIP

You choose the biggest power because you are looking for common multiples, so you need to take the maximum number of times the base appears.

4 120 expressed as a product of prime factors is 2 3 × 3 × 5

a Express 504 as a product of prime factors.

b Hence find:

i the highest common factor of 120 and 504

ii the lowest common multiple of 120 and 504.

5 There are 45 sweets and 72 chocolates in a bag. The sweets and chocolates are packed into gift bags such that each bag has the same number of sweets and chocolates. Find the largest number of gift bags that can be made.

6 Find the HCF and LCM of:

a 60, 75 and 90 b 48, 84 and 132 c 70, 210, 350.

7 Bus services A, B and C pass a particular bus stop every day. Bus service A passes the bus stop at 15 minute intervals, bus service B passes the bus stop at 20 minute intervals and bus service C passes the bus stop at 45 minute intervals. If all three buses are at the bus stop at 09:00, what is the next time that the three buses will all be at the bus stop?

8 When written as a products of prime factors

A = 2 3 × 3 3 × 7

B = 2 2 × 3 × 5 × 7

C = 2 × 5 2 × 7 3

Find:

a the HCF of A and C

b the LCM of B and C

d the LCM of A, B and C. Leave your answers in index notation.

c the HCF of A, B and C

1.5 Square roots and cube roots Square roots

The square root of a number is a number that when multiplied by itself gives that number.

For example the square of three is 3 2 = 3 × 3 = 9, so the square root of nine is √ 9 = √ 3 2 = 3

Notice that the power of 3 2 is divided by two when the square root is applied. The square root of a natural number n 2 is √ n 2 = √ n × n = n

Worked example 10 demonstrates how to use a product of prime factors to find the square root of a square number.

WORKED EXAMPLE 10

Find the square root of 144. Answer 144 = 2 4 ×

Cube roots

Express 144 as a product of prime factors.

To find the square root of 144, take the power of each base and divide each power by 2. Then multiply these numbers.

So the square root of 144 is 2 2 × 3 = 12

TIP

A square number will definitely have even powers in the product when expressed as the product of prime factors.

The cube root of a number is a number that when multiplied by itself twice gives that number.

For example eight is a cube number since 8 = 2 × 2 × 2 = 2 3. The cube root of eight is 3 √ 8 = 3 √ 2 3 = 2

Notice that the powers of two is divided by three when the cube root is applied. In general, 3 √ n 3 = 3 √ n × n × n = n

Worked example 11 demonstrates how to use a product of prime factors to find the cube root of a cube number.

WORKED EXAMPLE 11

Find the cube root of 5832. Answer 5832

Express 5832 as a product of prime factors.

To find the cube root of 5832, take the power of each base and divide each power by 3. Then multiply these numbers.

So the cube root of 5832 is 2 × 3 2 = 18

You are expected to remember the squares of numbers from 1 to 15 as well as their corresponding square roots. You should also remember the cubes of numbers from one to seven and their corresponding cube roots. You may even be asked to find other roots without the use of a calculator.

TIP

A cube number will have powers that are multiples of three when expressed as the product of prime factors.

WORKED EXAMPLE 12

Evaluate the following without the use of a calculator.

a 7 2 b √169 c 2 3 d 3 √64 e 5 √32

Answer

These are numbers you need to recall from memory.

a 7 2 = 7 × 7 = 49

b You need to know that 13 2 = 169 so √169 = 13.

c 23 = 2 × 2 × 2 = 8

d Since 4 × 4 × 4 = 64, 3 √64 = 4.

e 25 = 2 × 2 × 2 × 2 × 2 = 32. Hence, 5 √32 = 2.

WORKED EXAMPLE 13

a Express 108 as a product of prime factors.

b Given that h is a natural number and h ≠ 0, find h if 108 × h is a square number.

c Given that k is a natural number and k ≠ 0, find k if 108 × k is a cube number.

Answer

a 108 = 2 2 × 3 3

b 108 × h = 2 2 × 3 3 × 3

so that 108 × h = 2 2 × 3 4

∴ h = 3

c 108 × k = 2 2 × 3 3 × 2

so that 108 × k = 2 3 × 3 3

∴ k = 2

Use one of the methods from Worked example 7.

From the tip in Worked example 10, a square number will have even powers when it is expressed as a product of prime factors.

Base 2: 2 2 has an even power.

Base 3: 3 3 has an odd power.

For 108 × h to be a square number, the power of each prime base must be even, so you need to multiply 108 by 3 to make the power of base 3 even.

From the tip in Worked example 11, a cube number will have powers that are multiples of 3 when it is expressed as a product of prime factors.

Base 2: 2 2 has a power of 2. To make the power a multiple of 3, you must multiply it by 2.

Base 3: 3 3 has a power of 3.

For 108 × k to be a cube number, each power of its prime base must be a multiple of 3, so you need to multiply 108 by 2 to make the power of base 2 a 3.

Exercise 1.5

1 Without the use of a calculator, evaluate:

2 Find the value of:

3 Find the cube root of:

4 Without using a calculator, find the square root of each number. Leave your answers in index notation.

5 Without using a calculator, find the cube root of each number. Leave your answers in index notation.

6 Without using a calculator and leaving your answers in index form where appropriate, work out:

7 The square root of x is 2 2 × 7. Find the value of x

8 The cube root of y is 3 2 × 5 2. Find y. Leave your answer in index notation.

9 Without using a calculator and leaving your answer in index form work out 4 2 × 2 × √2

10 The numbers 252 and 1512, written as products of prime factors, are 252 = 2 2 × 3 2 × 7 1512 = 2 3 × 3 3 × 7

Find:

a the smallest positive integer value of h such that 252 × h is a square number

b the smallest integer value of k such that 1512 × k is a cube number.

1.6 Estimating and rounding

MATHEMATICS IN CONTEXT

Approximation and estimation are often required in real life.

For example, John wants to know the distance of a marathon. Ben says a marathon is approximately 42km, but Carl says a marathon is approximately 42.2km. Both Ben and Carl are estimating. Estimated values can be smaller or bigger than the actual value. When John checked using the internet, one website says a marathon is 42.195km, but another website says a marathon is 26.2 miles. When John uses an online converter to convert 26.2 miles it gives 42.16481km. 42.195km and 42.16481km are approximate values which are rounded from the actual value.

In this section, you will learn how to round and how to make estimations.

Rounding whole numbers

Worked example 14 demonstrates rounding numbers to the nearest tens, hundreds or thousands.

WORKED EXAMPLE 14

Round 23759 to the nearest:

a ten b hundred c thousand d ten thousand.

Answer

Locate the number in the column you are rounding to. If the digit to the right of it is zero, one, two, three or four, replace this digit with a zero as a place holder and leave the digit in the column unchanged. If the digit to the right of it is five, six , seven, eight or nine, replace this digit with a zero as a place holder and increase the digit in the column by one.

Step 1: identify the digit in the tens place.

Step 2: examine the digit to the right of the tens place.

So

23759 ≈ 23760

Use the approximate equal to sign ‘≈’ to show that you are approximating the original value of 23759 to 23760. An alternative way to indicate the approximation is to indicate the degree of accuracy you are rounding the number to:

23759 = 23760 (to the nearest ten).

You are rounding up the digit in the tens place by one.

b 2 3 7 5 9

ten thousands thousandshundreds tens ones

Step 1: identify the digit in the hundreds place.

Step 2: examine the digit to the right of the hundreds place using the same rule as in part a

Replace all digits to the right with a zero as a place holder.

So 23 759 = 23 800 (to the nearest hundred).

c 2 3 7 5 9

You are rounding up the digit in the hundreds place by one.

ten thousands thousandshundreds tens ones

Step 1: identify the digit in the thousands place.

Step 2: examine the digit to the right of the thousands place using the same rule as in part a

So 23 759 = 24 000 (to the nearest thousand).

Replace all digits to the right with a zero as a place holder.

You are rounding up the digit in the thousands place by 1.

d 2 3 7 5 9

ten thousands thousandshundreds tens ones

Step 1: identify the digit in the ten thousands place.

Step 2: examine the digit to the right of the ten thousands place using the same rule as in part a

Replace all digits to the right with a zero as a place holder.

So 23 759 = 20 000 (to the nearest ten thousand).

You are rounding up the digit in the ten thousands place by 1.

Rounding to decimal places

You can round a decimal to different numbers of decimal places. The method of rounding numbers to a specific number of decimal places is similar to the method shown for rounding whole numbers in Worked example 14.

For example: the place values in the number 1.234 5 are:

To round a number to a specific number of decimal places, look at the digit immediately to the right of the required decimal place.

If the digit to the right is five or more, round up the previous digit and remove all other digits after the specified decimal place.

If the digit to the right is four or less, remove the digits after the specified decimal place.

WORKED EXAMPLE 15

Round 3.456 12 to:

a three decimal places

Answers

b two decimal places

c one decimal place.

a To round 3.456 12 to three decimal places, look at the digit in the 4th decimal place.

place.

No rounding up required. Retain digits up to this decimal place.

3.456 12 = 3.456 (to three decimal places). Or 3.45612 = 3.456 (to 3 d.p.).

One is less than five, so remove all digits from here to the right.

WORKED EXAMPLE 15 CONTINUED

b To round 3.456 12 to two decimal places, look at the digit in the 3rd decimal place.

3.4 5 6 12

1st decimal place. 2nd decimal place.3rd decimal place.

Round up to six.Six is more than five, so round up the digit in the 2nd decimal place and remove all digits from here to the right.

3.45612 = 3.46 (to two decimal places). Or 3.45612 = 3.46 (to 2 d.p.).

c To round 3.456 12 to one decimal place, look at the digit in the 2nd decimal place.

3.4 5 612

1st decimal place.2nd decimal place. Round up to five. Five is equal to or more than five, so round up the digit in the 1st decimal place and remove all digits from here to the right.

3.45612 = 3.5 (to one decimal place). Or 3.45612 = 3.5 (to 1 d.p.).

Sometimes you will need to consider the context of a problem to decide which degree of accuracy to round the answer to.

For example, when rounding money, you would usually round the answer to two decimal places.

WORKED EXAMPLE 16

A website recorded 2300 views in the month of August. How many people visited the website each day?

Answers

There are 31 days in the month of August. To find the number of people who visited the website in a day, divide 2300 by 31.

2300 ÷ 31 = 74.193 548 39...

You need to round the answer to a reasonable degree of accuracy. Since the number of people visiting a website must be an integer, round the answer to the nearest whole number.

2300 ÷ 31 ≈ 74

You can say that approximately 74 people visited the website each day.

WORKED EXAMPLE 17

Alice buys 15 eggs for $4.70. How much does one egg cost?

Answers

To find the cost of one egg, divide $4.70 by 15.

$4.70 ÷ 15 = $0.3133333333...

Since dollars are made of cents, round the amount to the nearest cents (two decimal places).

$4.70 ÷ 15 ≈ $0.31

You can say that one egg cost approximately 31 cents.

Exercise 1.6

1 Round each number to the nearest whole number.

a 23.5 b 7.1 c 569.89 d 50.09.

2 Round each number to the nearest 10.

a 3452 b 4478 c 899 d 1035.

3 Round each number to the nearest 100.

a 1344 b 3288 c 78999 d 5072.

4 Round each number to the number of decimal places specified in the brackets.

a 101.03 (1 d.p.) b 0.198 323 (2 d.p.)

c 4.996 8 (2 d.p.) a 0.001 906 (3 d.p.).

5 The length of a plant is 44.306 741 cm when measured with a precise instrument. Express this length rounded to:

a four decimal places

b two decimal places.

6 The height of a fence is 2.774 1 metres. Express the height rounded to:

a the nearest metre

b the nearest 0.1 metres

c the nearest 0.01 metres

7 A group of 138 students are going on an excursion to the zoo. The school requires that there is one teacher for every 16 students.

a How many teachers are required to go on the excursion with the students?

b If one excursion bus can take a total of 30 students, how many buses are needed for the trip?

8 A painter needs to paint 26 doors in a building. It takes approximately 120 ml for one coat of paint for a door. The painter needs to put two coats of paint on each door.

One tin holds 500 ml of paint. How many tins of paint must the painter buy?

1.7 Accuracy and estimating Rounding to significant figures

As well as rounding numbers to decimal places, you can also round numbers to a specified number of significant figures.

MATHEMATICS IN CONTEXT

Consider these situations:

An engineer measured the width of a road to be 723.581 cm or 7.235 81 metres. The engineer must give all measurements rounded to one decimal place. Should the engineer give the measurement as 723.6 cm or 7.2 metres? Which measurement will be more accurate? Does accuracy depend on the number of decimal places or the number of digits?

This example shows that the degree of accuracy does not depend on the number of decimal places or digits. The degree of accuracy depends on the number of important or significant digits.

For example, 723.6 has four significant figures, but 7.2 has only two significant figures.

In general, there are five rules to identify significant figures in numbers.

A The first non-zero digit starting from the left is the first significant figure in a number.

B All non-zero digits are significant.

C All zeroes between non-zero digits are significant.

D In decimals numbers, the z eroes at the end are significant.

E In natural numbers, the zeroes at the end are generally not counted as significant. For example:

0.003160

Zeroes so not signi cant.

230500

4th signi cant gure, zeroes at the end of a decimal is signi cant.

3rd signi cant gure.

2nd signi cant gure.

First non-zero digit so this is the 1st signi cant gure.

Significant figures are digits in a number that contribute to its accuracy. A number is more accurate when it has more significant figures.

TIP

4th signi cant gure.

Zeroes at the end of natural numbers are not signi cant. 3rd signi cant gure, zero between non-zero digits are signi cant.

2nd signi cant gure.

1st signi cant gure.

There may be exceptions to rule E. If a number has been rounded, the zeroes at the end of the natural number may be significant depending on how many significant figures the number has been rounded to. For example, 4000 could be given to one significant figure, or two significant figures or three significant figures.

TIP

WORKED EXAMPLE 18

How many significant figures does each number have?

a 0.044 79 b 3.036 c 20.0 d 25 000

Answers

Use the five given rules to identify the number of significant figures for each number.

a 0.04479

4th signi cant gure.

3rd signi cant gure.

2nd signi cant gure.

1st signi cant gure.

Four significant figures.

b 3.036

4th signi cant gure.

3rd signi cant gure.

2nd signi cant gure.

1st signi cant gure.

Four significant figures.

c 20.0

3rd signi cant gure.

2nd signi cant gure.

1st signi cant gure.

Three significant figures.

d 25 000

Rule A states that the first non-zero digit starting from the left is the first significant figure.

Rule B states that all non-zero digits are significant.

Rule C states that all zeroes between non-zero digits are significant.

According to rule D, the zeroes at the end of a decimal are significant.

According to rule E, the zeroes at the end of natural numbers are generally not counted as significant unless the number is a rounded off value.

25 000 is the given value so you can take the value as it is.

2nd signi cant gure.

1st signi cant gure.

Two significant figures.

WORKED EXAMPLE 19

1 Round the number 0.006 045 to:

a three significant figures

2 Round the number 125 304 to:

a three significant figures

TIP

b two significant figures

b two significant figures

The method for rounding a natural number to a specific number of significant figures is similar to the method for rounding whole numbers. You need to use the digit zero as a place holder to maintain the value of each digit.

Answers

1 a 0.006 045

4th signi cant gure is ve so round up the 3rd signi cant gure from four to ve and remove all digits to the right.

3rd signi cant gure.

2nd signi cant gure.

1st signi cant gure.

0.006 045 = 0.006 05 (to three significant figures).

Or 0.006 045 = 0.006 05 (to 3 s.f.).

b 0.006045

4th signi cant gure.

3rd signi cant gure is four.Do not round up the previous digit. Remove all digits from here to the right.

2nd signi cant gure.

1st signi cant gure.

0.006 045 = 0.006 0 (to two significant figures).

Or 0.006 045 = 0.006 0 (to 2 s.f.).

To round a decimal to three significant figures, look at the fourth significant figure.

If the fourth significant figure is five or above, round up the third significant figure up by one and remove all other digits from the fourth significant figure to the right.

If the fourth significant figure is four or less, remove all digits to the right of the third significant figure.

To round a decimal to two significant figures, look at the third significant figure. If the third significant figure is five or above, round up the second significant figure by one and remove all other digits from the third significant figure to the right.

If the third significant figure is four or less, remove all digits to the right of the second significant figure.

2 a 125 304

4th signi cant gure is three so just replace all digits from here to the right with zero as a place holder.

3rd signi cant gure.

2nd signi cant gure.

1st signi cant gure.

125 304 = 125 000 (to three significant figures). Or 125 304 = 125 000 (to 3 s.f.).

b 1253 04

3rd signi cant gure is ve. Round up the digit in the 2nd signi cant gure from two to three and replace all digits from here to the right with zero as a.

2nd signi cant gure.

1st signi cant gure.

1253 04 = 130 000 (to two significant figures). Or 125 304 = 130 000 (to 2 s.f.).

Estimating

To round a natural number to three significant figures, look at the fourth significant figure.

If the fourth significant figure is five or above, round up the third significant figure by one and replace all other digits from the fourth significant figure to the right with zero as a place holder.

If the fourth significant figure is four or less, just replace all digits to the right of the third significant figure to the right with zero.

To round a natural number to two significant figures, look at the third significant figure. If the third significant figure is five or above, round up the second significant figure by one and replace all other digits from the third significant figure to the right with zero as a place holder.

If the third significant figure is four or less, just replace all digits from the third significant figure to the right with zero.

Rounding numbers enables you to estimate the answers to calculations. In general, you should not round values within a calculation and only round the final answer. However, if the question requires an estimation to n significant figures, round the numbers in the calculation to (n + 1) significant figures to estimate the answer. When estimating, you always need one significant figure more than the specified accuracy.

For example, if the question requires an estimation to one significant figure, round the numbers in the calculation to two significant figures to estimate the answer to the calculation.

WORKED EXAMPLE 20

Estimate the value of √38

Answer

To estimate the square root of a number, round the number in the square root to the nearest square number you recognize. In this case, the nearest square number is 36, so √38 ≈ √36 = 6

∴ √38 ≈ 6

WORKED EXAMPLE 21

Estimate the value of 8.995 × 10.09 1.958 correct to one significant figure.

Answer

To estimate the value of 8.995 × 10.09 1.958 to one significant figure, you need to round all numbers to two significant figures.

Use the approximately equal to sign ‘≈’ at this step because you are rounding the numbers in the original calculation.

Exercise 1.7

here because these values are exactly equal to the previous steps.

1 State the number of significant figures in each number.

2 Round each number to one significant figure.

12

3 Round each number to two significant figures.

4 Round each number, measurement or quantity to three significant figures.

a 423.1 b 0.982 45

d 12 345 e 249 788 metres

c 0.780 5

f 99 998 kg

5 Round each number, measurement or quantity to four significant figures.

a 2.812 5 b 0.012 345 6

d 6.999 5

e 434 901 apples

6 Round 3 418 726.5 to the nearest:

a three significant figures

c 120 586 ml

f 99 999

b whole number c million.

7 Estimate the answers to each calculation correct to one significant figure.

a 401.32 13.79 − 2. 09 2 a √ 401 + 10.1 14.99

c 63.51 10.005 × 0.798

8 Keiko wants to share a bag of 200 sweets with 12 friends. Keiko rounds to the nearest 10 and decides that 1 bag is enough to give each friend 20 sweets. Is she correct? Justify your answer.

9 Rishi wants to sew a pair of curtains and needs fabric that is 3.432 metres long and 1.654 metres wide. Using estimation, calculate:

a the total area of fabric needed for the curtains to one decimal place

b the total price of fabric needed if the cost is $8 per whole metre and the length is rounded to two decimal places

c the area of leftover fabric to three significant figures if the fabric Rishi buys is 2 metres wide.

SUMMARY

Are you able to...

?

find the reciprocal of a number

express numbers as a product of prime factors

find the LCM and HCF of two numbers

round values to specific numbers of decimal places or significant figures

estimate calculations by rounding numbers to a reasonable degree of accuracy convert between numbers and words

identify natural numbers, integers, prime numbers, square numbers, cube numbers, triangle numbers and rational and irrational numbers

calculate with squares, square roots, cubes and cube roots of numbers calculate with other powers and roots of numbers.

Past paper questions

1 Students in a college carry out a science experiment.

a When the results were posted online, there were 1279 views in the first day. Write 1279 correct to the nearest 10. [1]

b By the end of the week, there had been 15503 views. Write 15503 in words. [1]

Cambridge IGCSE International Mathematics 0607 Paper 32 Q3 c, d(i) March 2022

2 a Write 5249.6 correct to two significant figures. [1]

b Write 0.0030626 correct to three decimal places. [1]

Cambridge IGCSE International Mathematics 0607 Paper 21 Q2 June 2021

3 Write the number 25.0467

a correct to 1 decimal place, [1]

b correct to 3 significant figures, [1]

c correct to the nearest 10, [1]

d correct to the nearest 0.001. [1]

Cambridge IGCSE International Mathematics 0607 Paper 41 Q2a part (i) to (iv) November 2020

4 a Write 260512 correct to 3 significant figures. [1]

b Calculate √ 27 2 6 × 31 0.3 . Give your answer correct to 1 decimal place. [2]

Cambridge IGCSE International Mathematics 0607 Paper 42 Q2 a,c March 2021

5 a Write sixty thousand and three in figures. [1]

b Work out √ 729 [1]

c Write down all the factors of 10. [1]

d Write 965.384 correct to

i 1 decimal place, [1]

ii 3 significant figures, [1]

iii the nearest ten. [1]

Cambridge IGCSE International Mathematics 0607 Paper 31 Q1 a,b,d November 2021

6 Here is a list of numbers.

−2 √ 3 0.24 9

Write down one of the numbers from the list to complete each statement. You must use a different number in each statement.

.............................. is a natural number.

.............................. is an integer.

.............................. is a rational number. [3]

Cambridge IGCSE International Mathematics 0607 Paper 31 Q2b November 2021