preface chapter 1 prime numbers, factors and...

Chapter 1 Prime Numbers, Factors and Multiples

Secondary 1 (NA) Mathematics Tutorial 1A and 1B are designed to prepare Secondary 1 students in their understanding and application of mathematical concepts, skills and processes.

What’s covered in this book?Written in accordance with the latest syllabus, each chapter includes Objectives, Key Concepts and Formulae and Worked Examples to supplement and complement the lessons taught in school. Practice questions are structured as core, consolidation and challenging to ensure steady improvement and quick mastery of concepts. Books 1A and 1B cover all the topics for the entire school year.

Additional FeatureTeacher’s Desk that provides important notes, tips, examples and common student errors. Important concepts are highlighted to enhance understanding and retention. This encourages students to self-study, regardless of the level of competency they are at.

Answer KeyFully worked solutions are provided for students to understand better how each problem is solved. These also serve as a tool for self study and assessment.

There are 4 assessment papers: Mid Year Examination Paper 1 and 2 (found in Book 1A) and End of Year Examination Paper 1 and 2 (found in Book 1B).These assessment papers are available online, FREE.

This book will help students gain mastery and confidence in learning Mathematics systematically. The questions are also closely aligned to examinations in Singapore and hence students will be better equipped to face and excel in the examinations.

The Editorial Team

Preface

resources availablewww.onlineresources.sapgrp.com

EnhancedLearning resources availablewww.onlineresources.sapgrp.com

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Enhanced Learning

Secondary 1 (NA) Mathematics Tut2 2 1/24/2017 3:45:32 PM

Chapter 1 Prime Numbers, Factors and Multiples

Chapter Objectives Page

1

Prime Numbers, Factors and Multiples Recognise prime numbers Perform prime factorisation on a composite number Find the HCF and LCM of a group of numbers using prime

factorisation Find the square root and cube root of a number using prime

factorisation

1

2Real Numbers and Approximation Identify rational and irrational numbers Perform the four operations on real numbers Approximate using decimal places and significant figures Estimate the results of computations

23

3Introduction to Algebra Evaluate algebraic expressions and formulae Express real-world problems in algebraic terms

45

4Algebraic Manipulation Evaluate and simplify algebraic expressions Express real world problems in algebraic terms

63

5Solving Linear Equations Solve linear equations Formulate linear equations to solve word problems

81

6Number Patterns Recognise number patterns and find the terms of a sequence Find the general term of a sequence Solve problems involving number patterns and sequences

99

Fully worked solutions S1 – S23

Complete the course with Secondary 1 (NA) Mathematics Tutorial 1B (Chapters 7 – 12)

contents

Complete your learning with Mid Year Examination Paper 1 and 2.Download from www.onlineresources.sapgrp.com


Prime Numbers, Factors and Multiples

chaPter 1

Key Concepts and Formulae

1

2 3 × 3 × 3 × 5 × 5 when expressed in index notation is 33 × 52.

3 Prime Factorisation is the process of expressing a composite number as a product of its prime factors.

E.g. 504 = 23 × 32 × 7

4 Square and Square roots • Square of 6 = 62 = 36 ⇒ √

___ 36 = 6

• A perfect square is a number whose square root is a whole number. E.g. 1, 4, 9, 16, 25, 36 are perfect squares.

5 Cube and Cube roots • Cube of 5 = 53 = 125 ⇒ 3 √

____ 125 = 5

• A perfect cube is a number whose cube root is a whole number. E.g. 1, 8, 27, 64, 125 are perfect cubes.

6 Highest Common Factor (HCF) and Lowest Common Multiple (LCM) can be found using Prime Factorisation.

Prime Numbershas only 2 factors, 1 and itself

E.g. 2, 3, 5, 7, 11, ...

Composite Numbershas more than 2 factors

E.g. 4, 6, 8, 9, 10

Neither Prime nor Composite(0 and 1 only)

Whole numbersE.g. 0, 1, 2, 3, 4, 5, ...

ObjeCtives

Recognise prime numbers Perform prime factorisation on a composite number Find the HCF and LCM of a group of numbers using prime factorisation Find the square root and cube root of a number using prime factorisation

� � �Secondary 1 (NA) Mathematics Tutorial 1A© Singapore Asia Publishers Pte Ltd & Lim C. K. Chapter 1 Prime Numbers, Factors and Multiples


Teacher’s Desk

Only use prime numbers when finding the factors.

WorkeD example 1Find the prime factorisation of 150 in index notation.

Solution:

Method 1:

Using the factor tree:

150

×2 75

×3 25×2

×5 5×3×2

Hence, the prime factorisation of 150 is 2 × 3 × 52.

Method 2:

Using successive division:

1 5 027 532 55551

Hence, the prime factorisation of 150 is 2 × 3 × 52.

� � �Chapter 1 Prime Numbers, Factors and Multiples

Secondary 1 (NA) Mathematics Tutorial 1A© Singapore Asia Publishers Pte Ltd & Lim C. K.


WorkeD example 2Find the HCF and LCM of the numbers 40, 60 and 100.

Solution:

Method 1:

Using prime factorisation:

40 = 23 × 5

60 = 22 × 3 × 5

100 = 22 × 52

HCF = 22 × 5 → Extract common factors with lowest index.

LCM = 23 × 3 × 52 →Extract common factors with highest index and all remaining factors.

∴ HCF = 22 × 5 = 20 LCM = 23 × 3 × 52 = 600

Method 2:

Using successive division:

40, 60, 100220, 30, 50210, 15, 255 2 , 3 , 5

Stop dividing when there are no common factors between any two numbers

Common prime factors

∴ HCF = 2 × 2 × 5 = 20 LCM = 2 × 2 × 5 × 2 × 3 × 5 = 600

Teacher’s Desk

Use only prime numbers when performing successive division.

Teacher’s Desk

In Method 1, you will need to find the prime factorisation of each number first.




WorkeD example 3Buses on 3 different routes start their journey from the interchange at regular intervals.Buses for route A leave the interchange every 18 minutes, buses for route B leave every 20 minutes and buses for route C leave every 24 minutes. Given that all the first buses for all 3 routes leave at 7 a.m., what time will they next leave together?

Solution:

Find the LCM of 18, 20 and 24.

18, 20, 242 9 , 10, 122 9 , 5 , 62 9 , 5 , 3 3 , 5 , 13 1 , 5 , 15 1 , 1 , 1

3

LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360 360 minutes = 3 hours The buses will next leave together at 10 a.m.

WorkeD example 4Using prime factorisation, find the(a) square root of 784,(b) cube root of 216.

Solution:

(a) Using prime factorisation, 784 = 24 × 72

√____

784 = √_______________

(22 × 7) × (22 × 7) = 22 × 7 = 28

(b) Using prime factorisation, 216 = 23 × 33

3 √____

216 = 3 √_____________________

(2 × 3) × (2 × 3) × (2 × 3) = 2 × 3 = 6

Teacher’s Desk

Successive division is a more efficient way to find LCM/HCF.

Teacher’s Desk

√_____

a × a = a

3 √________

b × b × b = b




Core practice

Prime Factorisation1 Express the following numbers in index notation.

Teacher’s Desk

The index shows the number of times a base is multipled by itself.

53 = 5 × 5 × 5index

Base

(a) 2 × 2 × 3 × 3 × 7 (b) 2 × 5 × 5 × 5 × 5 × 5 × 11 × 11

(c) 7 × 7 × 11 × 23 × 23 × 23 (d) 3 × 3 × 3 × 3 × 3 × 37

(e) 3 × 3 × 7 × 7 × 7 × 13 (f) 2 × 11 × 11 × 11 × 19 × 19

(g) 5 × 7 × 17 × 17 × 17 × 17 (h) 3 × 3 × 3 × 23 × 23 × 29 × 29 × 29

(i) 11 × 13 × 19 × 19 × 13 × 13 (j) 5 × 19 × 3 × 5 × 3 × 3 × 19



2 Find the prime factorisation of the following numbers.

Teacher’s Desk

Methods to find the prime factorisation of a number Factor Tree Successive Division 12 2 12 3 4 2 6 2 2 3 3 1

∴ 12 = 22 × 3 ∴ 12 = 22 × 3

(a) 200 (b) 945

(c) 735 (d) 1000

(e) 1350 (f) 1372



1 (a) List all the factors of 48.

(b) List all the factors of 90.

(c) (i) Hence find all the common factors of 48 and 90. (ii) Hence state the HCF of 48 and 90.

2 (a) Identify the composite number from the list below and find its prime factorisation.

28, 17, 7

(b) Hence, express in index notation the product 28 × 17 × 7.

3 The volume of a dice in the shape of a cube is 3 375 mm2. Find the length of each side of the cube using prime factorisation.

Teacher’s Desk

(ii) Use your answer in (i) to identify the HCF.

Teacher’s Desk

A composite number is a whole number greater than 1 that has more than two factors.

Teacher’s Desk

Length of a cube

= 3 √_______

Volume

Consolidation practice

� �� Chapter 1 Prime Numbers, Factors and Multiples



RationalAll numbers that can be expressed as a __ b ,

where a and b are integers

Real NumbersAll rational and irrational numbers

IrrationalAll numbers that are not

rational

Non-terminating and non-recurring decimals

E.g.: π, √__

2 , 1 ___ √

__ 5

FractionsE.g.: 2 __ 3 , – 1 __ 5

IntegersE.g.: –3, –1, 0, 4, 9

Terminating/Recurring decimals

E.g.: 7.182, 2.3

Real Numbers and Approximation

chaPter 2

ObjeCtives

Identify rational and irrational numbers Perform the four operations on real numbers Approximate using decimal places and significant figures Estimate the results of computations

Key Concepts and Formulae

1

2 Operations on Integers

Addition and Subtraction

• a + (–b) = a – b• a – (–b) = a + b

Multiplication and Division

• a × (–b) = –ab• (–a) × (–b) = ab• a ÷ (–b) = – a __ b

• (–a) ÷ (–b) = a __ b

3 Significant Figures Generally, we count significant figures (s.f.) from the first non-zero number from the left. For example: (a) 3.015 has 4 s.f. (b) 0.047 has 2 s.f. (c) 100 378 has 6 s.f.

� �� Chapter 2 Real Numbers and Approximation



Teacher’s Desk

Integers include negative integers, positive integers and zero.

Teacher’s Desk

(–3)2 = 9

Teacher’s Desk

Order of operations: • Brackets • Power and roots • Multiplication and

Division (from left to right)

• Addition and Subtraction (from left to right)

Teacher’s Desk

Note how ‘=’ and ‘≈’ are used:139.863 ≈ 140but139.863 = 140 (correct to nearest whole number)

WorkeD example 1Consider the 8 numbers: 3.14, 3 √

____ –27 , �, 13, √

__ 3 , 3 __ 4 , 55.5 and 0.

Identify all the(a) integers,(b) rational numbers.

Solution:

(a) Integers: 3 √____

–27 , 13, 0 (Note that 3 √____

–27 = –3)

(b) Rational numbers: 3.14, 3 √____

–27 , 13, 3 __ 4 , 55.5, 0

WorkeD example 2Evaluate each of the following.(a) (–3) + 16 ÷ (24 – 8)(b) 5 – {6 × [(–3)2 – 2] ÷ 3}

Solution:

(a) (–3) + 16 ÷ (24 – 8) = (–3) + 16 ÷ 8 (perform operation in brackets first)= (–3) + 2 (perform division first)= –1

(b) 5 – {6 × [(–3)2 – 2] ÷ 3}= 5 – {6 × [7] ÷ 3} (perform operation in brackets first)

= 5 – {42 ÷ 3} (perform operation from left to right with × and ÷)

= 5 – 14= –9

WorkeD example 3Round off 139.863 correct to(a) the nearest whole number,(b) 1 decimal place,(c) 5 significant figures.

Solution:(a) 139.863 = 140 (correct to nearest whole number)

(b) 139.863 = 139.9 (correct to 1 d.p.)

(c) 139.863 = 139.86 (correct to 5 s.f.)

5 or more

5 or more

less than 5

� �� Chapter 2 Real Numbers and Approximation



� S� �Chapter 1


Chapter 1

Core Practice1. (a) 22 × 32 × 7 (b) 2 × 55 × 112

(c) 72 × 11 × 233

(d) 35 × 37 (e) 32 × 73 × 13 (f) 2 × 113 × 192

(g) 5 × 7 × 174

(h) 33 × 232 × 293

(i) 11 × 133 × 192

(j) 33 × 52 × 192

2. (a) 200100502551

22255

200 = 23 × 52

(b) 9453151053571

33357

945 = 33 × 5 × 7

(c) 7352454971

3577

735 = 3 × 5 × 72

(d) 10005002501252551

222555

1000 = 23 × 53

(e) 1350675225752551

233355

1350 = 2 × 33 × 52

(f) 13726863434971

22777

1372 = 22 × 73

3. (a) 50, 75 10, 15 2, 3

55

HCF = 52

= 25

(b) 24, 36 12, 18 6, 9 2, 3

223

HCF = 22 × 3 = 12

(c) 70, 85 14, 17

5 HCF = 5

(d) 84, 56 42, 2821, 14 3, 2

227

HCF = 22 × 7 = 28

(e) 42, 140, 16821, 70, 84 3, 10, 12

27

HCF = 2 × 7 = 14

(f) 20, 30, 6010, 15, 30 2, 3, 6

25

HCF = 2 × 5 = 10

(g) 100, 140, 280 50, 70, 140 25, 35, 70 5, 7, 14

225

HCF = 22 × 5 = 20

(h) 65, 108, 4251 HCF = 1

Note: Since the HCF of 65, 108 and 425 is 1, the three numbers are ‘coprime’ to each other.

4. (a) 84 = 22 × 3 × 7 396 = 22 × 32 × 11HCF = 22 × 3

∴ HCF = 22 × 3 = 12


preface chapter 1 prime numbers, factors and...

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