Higher Level Mathematics 9 781782 944379 ISBN 978 1 78294 437 9 MHN43 £19.99 (Retail Price) www.cgpbooks.co.uk CGP Mathematics for GCSE & IGCSE ® 8 cm 16 cm 12 cm 8 cm 16 cm 12 cm CGP Mathematics for GCSE & IGCSE ® P P CG CG Mathematics for GCSE & IGCSE® Higher Level for GCSE & IGCSE ® For the Grade 9-1 Course Thousands of questions for GCSE and IGCSE ® Maths! This indispensable CGP book is bursting with all the practice students need... • GCSE and IGCSE® Maths in one book... We’ve covered all the latest Grade 9-1 courses! • Easy-to-follow worked examples... Reminding students how it’s done — step-by-step! • Answers included at the back... So checking and marking is a breeze! It’s so useful, choosing another Maths book just wouldn’t add up ☻ P.S. This book is also available as an Online Edition that students can read on a PC, Mac or tablet — perfect for homework! P P CG CG OK, maybe not your life. But when it comes to exam revision, CGP are the undisputed champions. 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Mathematics GCSE & IGCSE for ® P P CG CG Editors: Rob Harrison, Shaun Harrogate, Paul Jordin, Sarah Oxley, Andy Park, David Ryan, Jonathan Wray Contributors: Katharine Brown, Eva Cowlishaw, Alastair Duncombe, Stephen Green, Philip Hale, Phil Harvey, Judy Hornigold, Claire Jackson, Mark Moody, Charlotte O’Brien, Rosemary Rogers, Manpreet Sambhi, Neil Saunders, Jan Walker, Kieran Wardell, Jeanette Whiteman Published by CGP MHN43 ~ 0219 - 18701 Clipart from Corel® Text, design, layout and original illustrations © Coordination Group Publications Ltd. (CGP) 2016 All rights reserved. This book is not endorsed by Cambridge International Examinations. ® IGCSE is a registered trademark of Cambridge International Examinations. 0800 1712 712 • www.cgpbooks.co.uk Contents Number, Ratio and Algebra Section 1 — Arithmetic, Multiples and Factors 1.1 Calculations ................................................... 2 1.2 Multiples and Factors ..................................... 5 1.3 Prime Numbers and Prime Factors ................ 7 1.4 LCM and HCF ............................................... 9 Section 2 — Approximations 2.1 Rounding ...................................................... 12 2.2 Upper and Lower Bounds ............................ 14 Section 3 — Fractions 3.1 Equivalent Fractions .................................... 17 3.2 Mixed Numbers ........................................... 18 3.3 Ordering Fractions ....................................... 19 3.4 Adding and Subtracting Fractions ............... 21 3.5 Multiplying and Dividing Fractions ............ 22 3.6 Fractions and Decimals ............................... 24 3.7 Fractions Problems ...................................... 27 Section 4 — Ratio and Proportion 4.1 Ratios ........................................................... 29 4.2 Dividing in a Given Ratio ............................ 32 4.3 More Ratio Problems ................................... 34 4.4 Proportion .................................................... 35 4.5 Ratio and Proportion Problems ................... 38 Section 5 — Percentages 5.1 Percentages ................................................... 40 5.2 Percentages, Fractions and Decimals .......... 42 5.3 Percentage Increase and Decrease ............... 44 5.4 Compound Percentage Change .................... 46 5.5 Percentages Problems .................................. 49 Section 6 — Expressions 6.1 Simplifying Expressions .............................. 51 6.2 Expanding Brackets ..................................... 52 6.3 Factorising — Common Factors .................. 56 6.4 Factorising — Quadratics ............................ 57 6.5 Algebraic Fractions ...................................... 59 6.6 Expressions Problems .................................. 63 Section 7 — Powers and Roots 7.1 Squares, Cubes and Roots .......................... 65 7.2 Indices and Index Laws .............................. 66 7.3 Standard Form ............................................ 71 7.4 Surds ........................................................... 73 Section 8 — Formulas 8.1 Writing Formulas ........................................ 77 8.2 Substituting into a Formula ........................ 78 8.3 Rearranging Formulas ................................ 80 8.4 Formulas Problems ..................................... 81 Section 9 — Equations 9.1 Solving Equations ....................................... 83 9.2 Writing Equations ....................................... 85 9.3 Iterative Methods ........................................ 87 9.4 Equations Problems .................................... 89 9.5 Identities...................................................... 91 Section 10 — Direct and Inverse Proportion 10.1 Direct Proportion ....................................... 92 10.2 Inverse Proportion ..................................... 94 10.3 Direct and Inverse Proportion Problems .................................................... 96 Section 11 — Quadratic Equations 11.1 Solving Quadratics by Factorising ............ 98 11.2 Completing the Square .............................100 11.3 The Quadratic Formula ............................102 11.4 Quadratic Equations Problems .................104 Section 12 — Simultaneous Equations 12.1 Simultaneous Equations ...........................106 12.2 More Simultaneous Equations .................108 Section 13 — Inequalities 13.1 Inequalities ...............................................110 13.2 Quadratic Inequalities ..............................111 13.3 Graphing Inequalities ...............................113 13.4 Linear Programming ................................115 Section 14 — Sequences 14.1 Term to Term Rules ..................................118 14.2 Using the nth Term ...................................121 14.3 Finding the nth Term ................................123 14.4 Arithmetic Series ......................................128 Section 15 — Straight-Line Graphs 15.1 Straight-Line Graphs ................................130 15.2 Gradients ..................................................132 15.3 Equations of Straight-Line Graphs ...........134 15.4 Parallel and Perpendicular Lines ..............136 15.5 Line Segments ..........................................138 15.6 Straight-Line Graphs Problems ................141 Section 16 — Other Types of Graph 16.1 Quadratic Graphs .....................................143 16.2 Cubic Graphs ............................................145 16.3 Reciprocal Graphs ....................................147 16.4 More Reciprocal Graphs ...........................148 16.5 Exponential Graphs ..................................150 16.6 Circle Graphs ............................................151 16.7 Trigonometric Graphs ..............................152 16.8 Transforming Graphs ................................154 16.9 Graphs Problems ......................................159 Section 17 — Using Graphs 17.1 Interpreting Real-Life Graphs ..................162 17.2 Drawing Real-Life Graphs .......................164 17.3 Solving Simultaneous Equations Graphically ...............................................166 17.4 Solving Quadratic Equations Graphically ...............................................168 17.5 Sketching Quadratic Graphs ....................170 17.6 Gradients of Curves ..................................171 Section 18 — Functions 18.1 Functions ..................................................173 18.2 Composite Functions ................................176 18.3 Inverse Functions ......................................177 18.4 Functions Problems ..................................178 Section 19 — Differentiation 19.1 Differentiating Powers of x ......................180 19.2 Finding Gradients .....................................181 19.3 Maximum and Minimum Points ..............183 19.4 Using Differentiation ................................185 Section 20 — Matrices 20.1 Matrix Addition and Subtraction .............188 20.2 Matrix Multiplication ...............................189 20.3 Inverse Matrices and Determinants .........192 Section 21 — Sets 21.1 Sets ........................................................... 195 21.2 Venn Diagrams ........................................ 196 21.3 Unions and Intersections ........................ 200 21.4 Complement of a Set .............................. 202 21.5 Subsets .................................................... 204 21.6 Sets Problems ......................................... 206 Geometry and Measures Section 22 — Angles and 2D Shapes 22.1 Angles and Lines ..................................... 208 22.2 Triangles ...................................................211 22.3 Quadrilaterals ...........................................213 22.4 Polygons ................................................... 215 22.5 Symmetry ................................................ 218 22.6 Angles and 2D Shapes Problems ............. 219 Section 23 — Circle Geometry 23.1 Circle Theorems 1 ................................... 221 23.2 Circle Theorems 2 .................................. 223 23.3 Circle Theorems 3 ................................... 225 23.4 Circle Geometry Problems ...................... 228 Section 24 — Units, Measuring and Estimating 24.1 Converting Metric Units — Length, Mass and Volume .................. 229 24.2 Converting Metric Units — Area and Volume ................................ 230 24.3 Metric and Imperial Units ........................231 24.4 Estimating in Real Life ............................232 Section 25 — Compound Measures 25.1 Compound Measures ................................233 25.2 Distance-Time Graphs ............................ 236 25.3 Velocity-Time Graphs ............................. 238 Section 26 — Constructions 26.1 Scale Drawings ....................................... 242 26.2 Bearings .................................................. 244 26.3 Constructions .......................................... 246 26.4 Loci .......................................................... 252 Section 27 — Pythagoras and Trigonometry 27.1 Pythagoras’ Theorem .............................. 254 27.2 Pythagoras’ Theorem in 3D ................... 256 27.3 Trigonometry — Sin, Cos and Tan ......... 258 27.4 The Sine and Cosine Rules ...................... 262 27.5 Sin, Cos and Tan of Larger Angles ........ 266 27.6 Trigonometry in 3D ................................. 269 27.7 Pythagoras and Trigonometry Problems ...................................................270 Section 28 — Vectors 28.1 Vectors and Scalars ..................................273 28.2 Magnitude of Vectors .............................. 276 28.3 Vectors Problems ..................................... 278 Section 29 — Perimeter and Area 29.1 Triangles and Quadrilaterals ................... 281 29.2 Circles and Sectors ................................. 284 29.3 Perimeter and Area Problems ................. 287 Section 30 — 3D Shapes 30.1 Plans, Elevations and Isometric Drawings ................................. 289 30.2 Volume ......................................................291 30.3 Nets and Surface Area ............................. 293 30.4 Spheres, Cones and Pyramids ................ 296 30.5 Rates of Flow ........................................... 298 30.6 Symmetry of 3D Shapes .......................... 299 30.7 3D Shapes Problems ................................ 300 Section 31 — Transformations 31.1 Reflections ............................................... 302 31.2 Rotations .................................................. 304 31.3 Translations .............................................. 307 31.4 Enlargements ........................................... 309 31.5 Combinations of Transformations ............313 31.6 Matrix Transformations .......................... 315 Section 32 — Congruence and Similarity 32.1 Congruence and Similarity ...................... 318 32.2 Areas of Similar Shapes .......................... 322 32.3 Volumes of Similar Shapes ..................... 323 32.4 Congruence and Similarity Problems ..... 324 Statistics and Probability Section 33 — Collecting Data 33.1 Using Different Types of Data ................ 326 33.2 Data-Collection Sheets and Questionnaires ......................................... 327 33.3 Sampling and Bias ................................... 331 Section 34 — Averages and Range 34.1 Averages and Range ................................. 335 34.2 Averages for Grouped Data ..................... 338 34.3 Interpreting Data Sets .............................. 339 Section 35 — Displaying Data 35.1 Tables and Charts .................................... 342 35.2 Stem and Leaf Diagrams ........................ 346 35.3 Frequency Polygons ................................ 348 35.4 Histograms ............................................... 349 35.5 Cumulative Frequency Diagrams ............ 352 35.6 Time Series .............................................. 355 35.7 Scatter Graphs ......................................... 358 35.8 Displaying Data Problems ....................... 361 35.9 Misrepresentation of Data ....................... 363 Section 36 — Probability 36.1 Calculating Probabilities ........................ 364 36.2 Listing Outcomes ................................... 366 36.3 Probability from Experiments ................ 368 36.4 The AND Rule for Independent Events ...................................371 36.5 The OR Rule ............................................ 372 36.6 Using the AND / OR Rules ..................... 374 36.7 Tree Diagrams ......................................... 375 36.8 Conditional Probability ............................ 377 36.9 Probability Problems .............................. 380 Answers .............................................................. 383 Index .................................................................. 448 Throughout the book, the more challenging questions are marked like this: 1 2 Section 1 — Arithmetic, Multiples and Factors Section 1 — Arithmetic, Multiples and Factors 1.1 Calculations Exercise 2 1 Work out the following without using your calculator. a) –4 + 3 b) –1 – 4 c) –12 + 15 d) 6 – 17 e) 4 – (–2) f) –6 – (–2) g) –5 + (–5) h) –5 – (–5) i) –23 – (–35) j) 48 + (–22) k) –27 + (–33) l) 61 – (–29) Example 2 Work out: a) 1 – (–4) b) –5 + (–2). a) 1 – (–4) = 1 + 4 = 5 b) –5 + (–2) = –5 – 2 = –7 Exercise 1 Answer these questions without using your calculator. 1 Work out the following. a) 5 + 1 × 3 b) 11 – 2 × 5 c) 18 – 10 ÷ 5 d) 24 ÷ 4 + 2 e) 35 ÷ 5 + 2 f) 36 – 12 ÷ 4 g) 2 × (4 + 10) h) (7 – 2) × 3 i) 4 + (48 ÷ 8) j) 56 ÷ (2 × 4) k) (3 + 2) × (9 – 4) l) (8 – 7) × (6 + 5) m) 2 × (8 + 4) – 7 n) 5 × 6 – 8 ÷ 2 o) 18 ÷ (9 – 12 ÷ 4) p) 100 ÷ (8 + 3 × 4) q) 7 + (10 – 9 ÷ 3) r) 20 – (5 × 3 + 2) s) 48 ÷ 3 – 7 × 2 t) 36 – (7 + 4 × 4) 2 Work out the following. a) 4 (5 3) 16 # - b) 15 3 8 2 ' + c) 6 3 2 4 (7 5) # # + + d) 7 5 6 (11 8) - + - e) 25 5 12 (9 5) ' ' - f) 5 6 7 8 2 4 # ' - + g) 21 (12 5) 3 3 ' # - h) 8 8 2 36 (11 2) ' ' - - Example 1 Work out: a) 20 – 12 ÷ 2 × 3 b) 30 ÷ (15 – 12) a) 20 – 12 ÷ 2 × 3 = 20 – 6 × 3 = 20 – 18 = 2 b) 30 ÷ (15 – 12) = 30 ÷ 3 = 10 BODMAS tells you the correct order to carry out mathematical operations. If there are two or more consecutive divisions and/or multiplications, they should be done in order, from left to right. The same goes for addition and subtraction. Brackets, Other, Division/Multiplication, Addition/Subtraction Order of Operations Negative Numbers Adding a negative number is the same as subtracting a positive number. So ‘+’ next to ‘–’ means subtract. Subtracting a negative number is the same as adding a positive number. So ‘–’ next to ‘–’ means add. 3 Section 1 — Arithmetic, Multiples and Factors Example 3 Work out: a) 24 ÷ (–6) b) (–5) × (–8). a) 24 ÷ (–6) = –4 b) (–5) × (–8) = 40 Exercise 3 Answer these questions without using your calculator. 1 Work out the following. a) 3 × (–4) b) (–15) ÷ (–3) c) 12 ÷ (–4) d) 2 × (–8) e) (–72) ÷ (–6) f) 56 ÷ (–8) g) (–16) × (–3) h) (–81) ÷ (–9) i) (–13) × (–3) j) 7 × (–6) k) 45 ÷ (–9) l) (–34) × 2 2 Work out the following. a) [(–3) × 7] ÷ (–21) b) [(–24) ÷ 8] ÷ 3 c) [55 ÷ (–11)] × (–9) d) [(–3) × (–5)] × (–6) e) [(–63) ÷ (–9)] × (–7) f) [35 ÷ (–7)] × (–8) g) [(–60) ÷ 12] × (–10) h) [(–12) × 3] × (–2) 3 Copy the following calculations and fill in the blanks. a) (–3) × = –6 b) (–14) ÷ = –2 c) × 4 = –16 d) ÷ (–2) = –5 e) (–8) × = –24 f) (–18) ÷ = 3 g) × (–3) = 36 h) ÷ 11 = –7 Example 4 Work out: a) 4.53 + 1.6 b) 8.5 – 3.07 1. Set out the sum by lining up the decimal points. 2. Fill in any gaps with 0’s. 3. Add or subtract the digits one column at a time. 8 . 5 0 – 3 . 0 7 5 . 4 3 4 1 b) a) 4 . 5 3 + 1 . 6 0 6 . 1 3 Decimals 1 Exercise 4 Answer these questions without using your calculator. 1 Work out the following. a) 2 + 1.8 b) 6 – 5.1 c) 12.74 + 7 d) 23 – 18.591 e) 5.1 + 1.8 f) 6.3 + 5.4 g) 11.7 – 8.2 h) 0.8 – 0.03 i) 10.83 + 7.4 j) 0.029 + 1.8 k) 91.7 + 0.492 l) 6.474 + 0.92 m) 67.5 – 4.31 n) 16.3 – 5.16 o) 9.241 – 2.8 p) 0.946 – 0.07 . 6 + 0 . 0 8 . 2 1 5 . 8 + . 4 6 . 4 0 6 . 7 5 + . 4 9 . 3 5 . 3 – 2 . 1 . 3 1 2 Copy the following calculations and fill in the blanks. a) b) c) d) When you multiply or divide two numbers which have the same sign, the answer is positive. When you multiply or divide two numbers which have opposite signs, the answer is negative. 4 Section 1 — Arithmetic, Multiples and Factors 3 Mo travels 2.3 km to the shops, then 4.6 km to town. How far has she travelled in total? 4 Sunita buys a hat for £18.50 and a bag for £31.99. How much does she spend altogether? 5 A block of wood is 4.2 m long. A 2.75 m long piece is cut from it. What length is left? 6 Jay’s meal costs £66.49. He uses a £15.25 off voucher. How much does he have left to pay? Exercise 6 Answer these questions without using your calculator. 1 Work out the following. a) 25.9 ÷ 10 b) 8.52 ÷ 4 c) 2.14 ÷ 4 d) 8.62 ÷ 5 e) 17.1 ÷ 6 f) 0.081 ÷ 9 g) 49.35 ÷ 7 h) 12.06 ÷ 8 Exercise 5 Answer these questions without using your calculator. 1 132 × 238 = 31 416. Use this information to work out the following. a) 13.2 × 238 b) 1.32 × 23.8 c) 1.32 × 0.238 d) 0.132 × 0.238 2 Work out the following. a) 0.92 × 10 b) 1.41 × 100 c) 72.5 × 1000 d) 16.7 × 8 e) 31.2 × 6 f) 68.8 × 3 g) 3.1 × 40 h) 0.7 × 600 i) 0.6 × 0.3 j) 0.05 × 0.04 k) 0.08 × 0.5 l) 0.04 × 0.02 m) 2.1 × 0.6 n) 8.1 × 0.5 o) 3.6 × 0.3 p) 1.6 × 0.04 q) 0.61 × 0.6 r) 5.2 × 0.09 s) 6.3 × 2.1 t) 1.4 × 2.3 u) 2.4 × 1.8 v) 3.9 × 8.3 w) 0.16 × 3.3 x) 0.64 × 0.42 3 1 litre is equal to 1.76 pints. What is 5 litres in pints? 4 1 mile is equal to 1.6 km. What is 3.5 miles in km? 5 Petrol costs £1.35 per litre. A car uses 9.2 litres during a journey. How much does this cost? 6 A shop sells apples for £1.18 per kg. How much would it cost to buy 2.5 kg of apples? Example 5 Work out 0.32 × 0.6 1. Multiply each decimal by a power of 10 to get a whole number multiplication. 2. Multiply the whole numbers. 3. Divide by the product of the powers of 10 you multiplied by in Step 1. 0.32 × 0.6 = 192 ÷ 1000 = 0.192 1 3 2 × 6 1 9 2 Example 6 Work out 0.516 ÷ 0.8 1. Multiply both numbers by 10, so you are dividing by a whole number. 2. Line up the decimal points to set out the calculation, then divide. So 0.516 ÷ 0.8 = 0.645 0.516 ÷ 0.8 = 5.16 ÷ 8 0.32 × 0.6 32 × 6 × 100 × 10 0. 6 4 5 8 5. 1 6 0 5 3 4 g 5 Section 1 — Arithmetic, Multiples and Factors Exercise 7 — Mixed Exercise Answer these questions without using your calculator. 1 At midday the temperature was 6 °C. By midnight, the temperature had decreased by 7 °C. What was the temperature at midnight? 2 Work out: –4.2 – (1.5 × –0.3) 3 Ashkan spends £71.42 at the supermarket. His receipt says that he has saved £11.79 on special offers. How much would he have spent if there had been no special offers? 4 On his first run, Ted sprints 100 m in 15.32 seconds. On his second run, he is 0.47 seconds quicker. How long did he take on his second run? 5 Asha bought 2 CDs each costing £11.95 and 3 CDs each costing £6.59. She paid with a £50 note. How much change did she receive? 6 It costs £31.85 to buy 7 identical DVDs. How much would it cost to buy 3 DVDs? 7 A single pack of salt and vinegar crisps costs 70p. A single pack of cheese and onion crisps costs 65p. A multipack of 3 salt and vinegar and 3 cheese and onion costs £3.19. How much would you save buying a multipack instead of the equivalent amount in individual packs? 2 3 5 1 . 2 Work out the following. a) 6.4 ÷ 0.2 b) 1.56 ÷ 0.2 c) 0.624 ÷ 0.3 d) 8.8 ÷ 0.2 e) 3.54 ÷ 0.4 f) 3.774 ÷ 0.4 g) 5.75 ÷ 0.5 h) 0.275 ÷ 0.5 i) 22.56 ÷ 0.03 j) 16.42 ÷ 0.02 k) 0.257 ÷ 0.05 l) 1.08 ÷ 0.08 m) 7.665 ÷ 0.03 n) 0.039 ÷ 0.06 o) 7.5 ÷ 0.05 p) 50.4 ÷ 0.07 q) 0.9 ÷ 0.03 r) 0.71 ÷ 0.002 s) 63 ÷ 0.09 t) 108 ÷ 0.4 u) 1.76 ÷ 0.008 v) 8.006 ÷ 0.2 w) 20.16 ÷ 0.007 x) 1.44 ÷ 1.2 3 A school jumble sale raises £412.86. The money is to be split equally between three charities. How much will each charity receive? 4 It costs £35.55 to buy nine identical books. How much does one book cost? 5 A 2.72 m ribbon is cut into equal pieces of length 0.08 m. How many pieces will there be? 6 It costs £6.93 to buy 3.5 kg of pears. How much do pears cost per kg? 1.2 Multiples and Factors A multiple of a number is a product of that number and any other number. A common multiple of two numbers is a multiple of both of those numbers. Multiples Example 1 a) List the multiples of 5 between 23 and 43. These are the numbers between 23 and 43 that 5 will divide into. b) Which of the numbers in the box on the right are common multiples of 2 and 7? 25, 30, 35, 40 24 7 28 42 35 The multiples of 2 are 24, 28 and 42, while the multiples of 7 are 7, 28, 35 and 42. 28, 42 6 Section 1 — Arithmetic, Multiples and Factors Exercise 1 1 List the first five multiples of: a) 9 b) 13 c) 16 2 a) List the multiples of 8 between 10 and 20. b) List the multiples of 12 between 20 and 100. c) List the multiples of 14 between 25 and 90. 3 Write down the numbers from the box that are: a) multiples of 10 b) multiples of 15 c) common multiples of 10 and 15 4 a) List the multiples of 3 between 19 and 35. b) List the multiples of 4 between 19 and 35. c) List the common multiples of 3 and 4 between 19 and 35. 5 List all the common multiples of 5 and 6 between 1 and 40. 6 List all the common multiples of 6, 8 and 10 between 1 and 100. 7 List all the common multiples of 9, 12 and 15 between 1 and 100. 8 List the first five common multiples of 3, 6 and 9. 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 Factors A number’s factors divide into it exactly. A common factor of two numbers is a factor of both of those numbers. Example 2 a) Write down all the factors of: (i) 18 (ii) 16 (i) 18 = 1 × 18 18 = 2 × 9 18 = 3 × 6 So the factors of 18 are 1, 2, 3, 6, 9, 18. (ii) 16 = 1 × 16 16 = 2 × 8 16 = 4 × 4 So the factors of 16 are 1, 2, 4, 8, 16. 1, 2 1. Check if 1, 2, 3... divide into the number. 2. Stop when a factor is repeated. These are the numbers which appear in both lists from part (a). Exercise 2 1 List all the factors of each of the following numbers. a) 10 b) 4 c) 13 d) 20 e) 25 f) 24 g) 35 h) 32 i) 40 j) 50 k) 9 l) 15 m) 36 n) 49 o) 48 2 a) Which number is a factor of all other numbers? b) Which two numbers are factors of all even numbers? c) Which two numbers must be factors of all numbers whose last digit is 5? d) Which four numbers must be factors of all numbers whose last digit is 0? b) Write down the common factors of 18 and 16. 7 Section 1 — Arithmetic, Multiples and Factors 3 A baker has 12 identical cakes. In how many different ways can he divide them up into equal packets? List the possibilities. 4 In how many different ways can 100 identical chairs be arranged in rows of equal length? List all the ways the chairs can be arranged. 5 a) List all the factors of: (i) 15 (ii) 21. b) Hence list the common factors of 15 and 21. 6 List the common factors of each of the following pairs of numbers. a) 15, 20 b) 12, 15 c) 30, 45 d) 50, 90 e) 25, 50 f) 24, 32 g) 36, 48 h) 64, 80 i) 45, 81 j) 96, 108 7 List the common factors of each of the following sets of numbers. a) 15, 20, 25 b) 12, 18, 20 c) 30, 45, 50 d) 15, 16, 17 e) 8, 12, 20 f) 9, 27, 36 g) 24, 48, 96 h) 33, 121, 154 Exercise 1 1 Consider the following list of numbers: 11, 13, 15, 17, 19 a) Which number in the list is not prime? b) Find two factors greater than 1 that can be multiplied together to give this number. 2 a) Which three numbers in the box on the right are not prime? b) Find two factors greater than 1 for each of your answers to (a). 3 Write down the prime numbers in this list: 5, 15, 22, 34, 47, 51, 59 4 a) Write down all the prime numbers less than 10. b) Find all the prime numbers between 20 and 50. 5 a) For each of the following, find a factor greater than 1 but less than the number itself. (i) 4 (ii) 14 (iii) 34 (iv) 74 b) Explain why any number with last digit 4 cannot be prime. 6 Without doing any calculations, explain how you can tell that none of the numbers in this list are prime. A prime number is a number that has no other factors except itself and 1. 1 is not a prime number. Example 1 Which of the numbers in the box on the right are prime? 16 17 18 19 20 16 = 2 × 8, 18 = 3 × 6, 20 = 4 × 5 17 has no factors other than 1 and 17. 19 has no factors other than 1 and 19. So the prime numbers are 17 and 19. 1. Look for factors of each of the numbers. 2. If there aren’t any, it’s prime. 31 33 35 37 39 20 30 40 50 70 90 110 130 1.3 Prime Numbers and Prime Factors 8 Section 1 — Arithmetic, Multiples and Factors Writing a Number as a Product of Prime Factors Whole numbers which are not prime can be broken down into prime factors. Example 2 Write 12 as the product of prime factors. Give your answer in index form. Make a factor tree. 1. Find any two factors of 12. Circle any that are prime. 2. Repeat step 1 for any factors which aren’t prime. 3. Stop when all the factor tree’s branches end in a prime. 4. Give any repeated factors as a power, e.g. 2 × 2 = 22 — this is what is meant by index form. 12 2 2 6 3 12 = 2 × 2 × 3 = 22 × 3 Exercise 2 Give your answers to these questions in index form where appropriate. 1 Write each of the following as the product of two prime factors. a) 14 b) 33 c) 10 d) 25 e) 55 f) 15 g) 21 h) 22 i) 35 j) 39 k) 77 l) 121 64 16 32 ? 8 ? ? ? ? ? 4 64 16 4 8 ? ? ? ? ? ? 4 64 8 8 ? ? ? ? ? ? 4 4 2 a) Copy and complete the three factor trees below. (i) (ii) (iii) b) Use each of your factor trees to write down the prime factors of 64. What do you notice? 3 Write each of the following as the product of prime factors. a) 6 b) 30 c) 42 d) 66 e) 70 f) 46 g) 110 h) 78 i) 190 j) 210 k) 138 l) 255 4 Write each of the following as the product of prime factors. a) 8 b) 44 c) 24 d) 16 e) 48 f) 72 g) 90 h) 18 i) 50 j) 28 k) 27 l) 60 m) 98 n) 36 o) 150 p) 132 q) 168 r) 225 s) 325 t) 1000 9 Section 1 — Arithmetic, Multiples and Factors Exercise 1 1 Find the LCM of each of the following pairs of numbers. a) 3 and 4 b) 3 and 5 c) 6 and 8 d) 2 and 10 e) 6 and 7 f) 4 and 9 g) 10 and 15 h) 15 and 20 2 Find the LCM of each of the following sets of numbers. a) 3, 6, 8 b) 2, 5, 6 c) 3, 5, 6 d) 4, 9, 12 e) 5, 7, 10 f) 5, 6, 9 3 Laurence and Naima are cycling around a circular course. They leave the start-line at the same time and need to do 10 laps. It takes Laurence 8 minutes to do one lap and Naima 12 minutes. a) After how many minutes does Laurence pass the start-line? Write down all possible answers. b) After how many minutes does Naima pass the start-line? Write down all possible answers. c) When will they first pass the start-line together? 4 Mike visits Oscar every 4 days, while Narinda visits Oscar every 5 days. If they both visited today, how many days will it be before they visit on the same day again? 5 A garden centre has between 95 and 205 potted plants. They can be arranged exactly in rows of 25 and exactly in rows of 30. How many plants are there? 6 There are between 240 and 300 decorated plates hanging on a wall, and the number of plates divides exactly by both 40 and 70. How many plates are there? 7 Jill divides a pile of sweets into 5 equal piles. Kay then divides the same sweets into 7 equal piles. What is the smallest number of sweets there could be? The least common multiple (LCM) of a set of numbers is the smallest of their common multiples. The highest common factor (HCF) of a set of numbers is the largest of their common factors. Example 1 Find the least common multiple (LCM) of 4, 6 and 8. Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28... Multiples of 6 are: 6, 12, 18, 24, 30, 36... Multiples of 8 are: 8, 16, 24, 32, 40, 48... So the LCM of 4, 6 and 8 is 24. 1. Find the multiples of 4, 6 and 8. 2. The LCM is the smallest number that appears in all three lists. 1.4 LCM and HCF 5 Square numbers have all their prime factors raised to even powers. For example, 36 = 22 × 32 and 64 = 26. a) Write 75 as a product of prime factors. b) What is the smallest number you could multiply 75 by to form a square number? Explain your answer. 6 By first writing each of the following as a product of prime factors, find the smallest integer that you could multiply each number by to give a square number. a) 180 b) 250 c) 416 d) 756 e) 1215