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preface Chapter 1 Prime Numbers, Factors and Multiples Secondary 1 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 feature Teacher’s Desk that provides important notes, tips, examples and common student errors. Important concepts are highlighted to enhance understanding and retention. Tests and Assessments 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). Answer Key Fully 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. It is hoped that this book will help students to gain confidence in the subject and be better equipped to face the examinations. The Editorial Team Contents 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 2 Real 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 3 Introduction to Algebra Evaluate algebraic expressions and formulae Express real-world problems in algebraic terms 44 4 Algebraic Manipulation Evaluate and simplify algebraic expressions Factorise algebraic expressions by extracting common factors Express real world problems in algebraic terms 61 5 Solving Linear Equations and Inequalities Solve linear equations Formulate linear equations to solve word problems 84 6 Number 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 112 7 Ratio, Rate and Speed Find ratios involving two or more quantities Calculate average speed Convert speed from one unit to another Solve problems involving ratio, rate and speed 130 Mid Year Examination Paper 1 156 Mid Year Examination Paper 2 162 Fully worked solutions Complete the course with Secondary 1 Mathematics Tutorial 1B (Chapters 8 – 14) S1 – S38 Chapter 1 Prime Numbers, Factors and Multiples 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 Key Concepts and Formulae 1 Whole numbers E.g. 0, 1, 2, 3, 4, 5, ... Neither Prime nor Composite (0 and 1 only) Prime Numbers has only 2 factors, 1 and itself E.g. 2, 3, 5, 7, 11, ... Composite Prime Numbers has more than 2 factors E.g. 4, 6, 8, 9, 10 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 ⇒ √ 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. ___ 3 © Singapore Asia Publishers Pte Ltd ____ Chapter 1 Prime Numbers, Factors and Multiples WORKeD eXAMPLe 1 Findtheprimefactorisationof150inindexnotation. Solution: Method 1: Usingthefactortree: 150 2 2 × 75 2 × 3 × 25 × 3 × 5 × Hence,theprimefactorisationof150is2×3×52. Method 2: Usingsuccessivedivision: 2 150 Teacher’s Desk 3 75 Onlyuseprimenumbers when finding the factors. 5 25 5 5 1 Hence,theprimefactorisationof150is2×3×52. ©SingaporeAsiaPublishersPteLtd 2 Chapter 1 Prime Numbers, Factors and Multiples 5 2 WORKeD eXAMPLe FindtheHCFandLCMofthenumbers40,60and100. Solution: Method 1: Teacher’s Desk InMethod1,youwill need to find the prime factorisationofeach number first. Usingprimefactorisation: 40 =23×5 60 =22×3×5 100 =22×52 HCF =22×5 →Extractcommonfactorswithlowest index. LCM =23×3×52 → Extractcommonfactorswithhighest indexandallremainingfactors. ∴HCF=22×5=20 LCM=23×3×52=600 Method 2: Usingsuccessivedivision: 2 40,60,100 Commonprime factors Teacher’s Desk 2 20,30,50 Stopdividingwhen therearenocommon factorsbetweenany twonumbers 5 10, 15,25 2, 3,5 ∴HCF=2×2×5=20 LCM=2×2×5×2×3×5=600 ©SingaporeAsiaPublishersPteLtd 3 Chapter 1 Prime Numbers, Factors and Multiples Useonlyprimenumbers whenperformingprime factorisation. WORKeD eXAMPLe 3 Buses on 3 different routes start their journey from the interchange at regularintervals. BusesforrouteAleavetheinterchangeevery18minutes,busesforroute Bleaveevery20minutesandbusesforrouteCleaveevery24minutes. Given that all the first buses for all 3 routes leave at 7 a.m., what time will theynextleavetogether? Solution: FindtheLCMof18,20and24. 2 18,20,24 Teacher’s Desk Successivedivisionis a more efficient way to find LCM/HCF. 2 9,10,12 2 9,5,6 3 9,5,3 3 3,5,1 5 1,5,1 1,1,1 LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360 360 minutes=3hours Thebuseswillnextleavetogetherat10a.m. WORKeD eXAMPLe 4 Using prime factorisation, find the (a) squarerootof784, (b) cuberootof216. Solution: Teacher’s Desk _____ =a √a × a ________ 3 =b √ b × b × b (a) Usingprimefactorisation,784=24×72 (b) Usingprimefactorisation,216=23×33 ©SingaporeAsiaPublishersPteLtd _______________ ____ 784=√(22×7)×(2 2×7) √ 2 =2 ×7 =28 3 ____ 3 _____________________ √ 216=√ (2×3)×(2×3)×(2×3) =2×3 =6 4 Chapter 1 Prime Numbers, Factors and Multiples core Practice Prime Factorisation 1 Expressthefollowingnumbersinindexnotation. Teacher’s Desk index 53=5×5×5 Base To find the prime factorisation of a number: Method 1:Factortree Method 2:Successivedivision (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 ©SingaporeAsiaPublishersPteLtd 5 Chapter 1 Prime Numbers, Factors and Multiples 2 Findtheprimefactorisationofthefollowingnumbers. Teacher’s Desk Anumberisdivisibleby3ifthesumof itsdigitsisdivisibleby3.Anumberis divisibleby5ifitslastdigitis0or5. (a) 200 (b) 945 (c) 735 (d) 1000 (e) 1350 (f) 1372 ©SingaporeAsiaPublishersPteLtd 6 Chapter 1 Prime Numbers, Factors and Multiples