repeated multiplication of a number, for example, 3 5. square roots and cube roots of numbers

NUMBER 456789012 Numbers form many interesting patterns. You already know about odd and even numbers. Pascal s triangle is a number pattern that looks like a triangle and contains number patterns. Fibonacci numbers are found in many living things: the number of petals on a flower will be a Fibonacci number. You will learn how these different patterns are formed to help you to understand how numbers behave. 890124 56789012 456789012 890 012456789 789012 890124 245678 4567890124567890124 124567890124 2456789012 4567890 45678901245678901245678901245678901245678 56 789012456789012 45678901245678 9012456789012456789012456789012456789012456789012 5678901245678901245678901245678901245678901245 567890124567890124567890 124567890124 5678 2456789012 456789012 012456789 2456789012456789012456789012456789012456 789012456789012 245 45678 901245 678901245678901245678901245678901245678901245678901245678 4 6789012456789012456789012456789012456789012456789012 24 678 78 901245 678901245678901245678 9012456789012456789012456789012456 789012456789012 456789012 9012456789012 0124567890 124567890124567890124 5678 0124567890124 567890124567890124567 890 12456789012456789 0124567890124567 89012456789012456789012456789012456789012456789012456789012456789012456789012456789012 456789012456789012456 7890124567890124567 890124 67890124567890 245678901 2456789012 890124 678901245678901 4567890124567890124567890124567 01245678901245678901245675678 9012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012 45678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678 90124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 12456789012456789012 4567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567 89012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012 4567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 12456789012456789012456789012456 78901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901 2456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456 7890124567890124567890124567890124567890124567890124567890124567890124567890124567890 124567890124567890124567890124567890124567890 1245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245 67890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 124567890124567890124567890124567890124567890124567890124567890124567890 124567890124567890124567890124567890124567890124567890124 56789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789 0124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124 567890124567890124567890124567890124567890124567890124567890 1245678901245678901245678901245678901245678901245678901245678901245678 9012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012 45678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678 90124567890124567890124567890124567890124567890 12456789012456789012456789012456789012456789012456789012456789012456789012456789012 4567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567 89012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012 4567890124567890124567890124567890 12456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456 78901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901 2456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456 7890124567890124567890 124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 124 48901

012456789 24 012456789 012456789 56789012 456789012 890124 012456789 124 In this chapter you will: 56789012 456789012 890124 Wordbank 012456789 9 5 5678901245678 56789 5678 45678901245678901245 5678901245678 124 identify special groups of numbers: triangular, 12456789012456789 composite number A number 5678 with more than 012456789 2456789012 4567890124 square, Fibonacci, Pascal s triangle and two factors. 6789012 45678901245678901245678901245678901245678 56789012 012456789 palindromes 789012456789012 45678901245678 divisibility test A rule for testing whether a 245678 901245678901245678901245678901245678901245678901245678 test 5678901245678901245678901245678901245678901245 numbers for divisibility number is divisible by a specific value, for 56789 567890124567890124567890 12456789012456789 24 identify the factors of a number 5678 and distinguish example, divisible by. 2456789012 4567890124 between prime and composite numbers 012456789 factor A value that divides evenly into a given 901456789012456789012456789012456 789012456789012 45678901 9012 901245 find 678901245678901245678901245678901245678901245678901245678901 the highest common factor of two or more number, for example, is a factor of 15. 9012456789012456789012456789012456789012456789012 45678 numbers 45678 901245 678901245678901245678 factor 9012456789012456789012456789012456 tree A diagram that lists the prime factors 78901245678 89012456789012 9012456789012 0124567890 124567890124567890124 5678 0124567890124 567 express 4567890 a 12456789012456789 number as a product 0124567890124567 of its prime factors of a number. 890124567890124567890124567890124567890124567890124567890124567890124 calculate squares 456789012456789012456 and cubes 7890124567890124567 index 890124 notation Using 67890124567890 powers to write the 245678901 2 90124567890124 4567890124 678901245678901 4567890124567890124567890124567 01245678901 456789012456789 estimate 56789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012 and calculate square roots and cube repeated multiplication of a number, for 9012456789012401245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678 45678901245678 roots example, 5. 567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 12 8901245678901290124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567 45678901245678 find 456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012 square roots and cube roots of numbers palindrome A number or word that reads the 789012456789012 expressed 90124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 as a product of their prime factors. same forward and backward, for example, 2002, 12456789012456 2456789012456745678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901 and madam. 78901245678901289012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456789012456 12456789012456 4567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 prime number A number with only 124567890124567890124567890 two factors, 67890124567890178901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245678901245 1 and the number itself. 1245678901245624567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890124567890 567890124567890 7890124567890124567890124567890124567890124567890124567890124567890124567890 124567890124567890124567890124567890124 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Start up Worksheet -01 Brainstarters Skillsheet -01 Classifying whole numbers 1 List the first ten even numbers. 2 Sort these numbers, putting all the even numbers in one group, and the odd numbers in another: 17 2002 71 14 60 02 2 748 691 90 704 006 1 95 1 2074 1 000 000 99 999 1256 Find all the numbers that divide into 6. 4 Find all the numbers that divide into 24. 5 Find all the even numbers that divide into 6. 6 Find all the odd numbers that divide into 90. 7 How can you tell if a number is even without dividing it? 8 How can you recognise an odd number? 9 Write the next three numbers in each of these patterns: a 8, 10, 12,,, b 27, 0,,,, c 101, 10, 105,,, d 9, 7, 5,,, e 44, 9, 4,,, f 7, 15, 2,,, 10 What is 8 squared? 11 What is 27? 12 Find two numbers that have a product of 48. Worksheet -02 Triangular and square numbers -01 Special number patterns The numbers 1, 2,, 4, 5, are called the counting numbers. There are groups of counting numbers which make special patterns. We will investigate some of them. Exercise -01 TLF L 196 1 Triangular numbers are shown in the diagram below. Circus towers: triangular towers 1 6 10 78 NEW CENTURY MATHS 7

a Why are they called triangular numbers? b Work out all the triangular numbers less than 100. c Complete four more lines of this pattern: 1 = 1 1 + 2 = 1 + 2 + = 6 1 + 2 + + 4 = 10 d Describe how the pattern in part c works. e Use what you have worked out to help you find the 100th triangular number. (Hint: Do you know a quick way to add up all the numbers from 1 to 100?) 2 Square numbers are shown in the diagram below. TLF L 195 Circus towers: square stacks 1 4 9 16 a Why are these called square numbers? b Work out all the square numbers up to 100. c Complete four more lines of this pattern: 1 = 1 1 + = 4 1 + + 5 = 9 1 + + 5 + 7 = 16 d Describe how the pattern works. e Work out another pattern to help you find the square numbers. What is the 50th square number? f Complete four more lines of these patterns: i 1 = 1 2 ii 2 2 = 1 2 + (1 + 2) 1 + 2 + 1 = 2 2 2 = 2 2 + (2 + ) 1 + 2 + + 2 + 1 = 2 4 2 = 2 + ( + 4) g Each square number is said to be the sum of two consecutive triangular numbers. + = + 6 = Show that this is true for the square numbers up to 100. h Find two numbers that are both triangular numbers and square numbers. 9 CHAPTER EXPLORING NUMBERS 79

Worksheet -0 Fibonacci numbers Leonardo Fibonacci was an Italian mathematician who lived in the early 1th century. He discovered this pattern when studying the breeding habits of rabbits: 1, 1, 2,, 5, 8, 1, 21,... The diagram below illustrates this. The vertical arrows labelled Birth indicate the new offspring of a pair of rabbits every two months. The unlabelled arrows indicate the same pair of rabbits. After each month, the number of pairs is a term in Fibonacci s pattern. Original pair of rabbits 1 pair 1st generation Birth 1 pair 2nd generation Birth 2 pairs rd generation Birth Birth pairs 4th generation 5 pairs a How is the Fibonacci pattern formed? b Add five more lines to this pattern: 1 1 1 + 1 = 2 1 + 2 = 2 + = 5 + 5 = 8 c Write the first 20 Fibonacci numbers. i Write every third Fibonacci number, beginning with 2. What number divides evenly into all these numbers? ii Write every fourth Fibonacci number, beginning with. What number divides evenly into all these numbers? iii Write every fifth Fibonacci number, beginning with 5. What number divides evenly into all these numbers? d i Find any triangular numbers in the Fibonacci numbers up to 100. ii Find any square numbers in the Fibonacci numbers up to 100. e Pairs of Fibonacci numbers are found by counting along the spirals on pine cones. Investigate how and where else Fibonacci numbers occur in nature. 80 NEW CENTURY MATHS 7

4 Blaise Pascal, a French mathematician who lived 1 in the 17th century, studied a triangle of 1 1 numbers known to the Chinese as the Yanghui 1 2 1 triangle. Each row of the triangle is created using 1 1 the numbers in the row above it. The triangle is 1 4 6 4 1 known as Pascal s triangle. The first seven rows 1 5 10 10 5 1 are shown at the right. 6 15 20 15 6 1 a Complete the next four rows of Pascal s triangle. b Describe how the pattern works. c Add each row in Pascal s triangle. What do you notice? d The diagonals in Pascal s triangle produce some interesting patterns. Write the triangular numbers using Pascal s triangle. e We can even find Fibonacci numbers in this pattern. Rewrite the triangle above as a rightangled triangle. Add along the arrows 1 to find the Fibonacci 1 1 numbers. 1 2 1 1 1 1 4 6 4 1 1 5 10 10 5 1 5 A palindrome is a word, number or sentence that reads the same forward and backward. The following number, words and sentence are all palindromes: noon 151 Able was I ere I saw Elba (Napoleon Bonaparte) Worksheet -04 Pascal s triangle Place names can be palindromes. a Select the palindromes from these numbers. 447 7 656 281 7 22 899 191 797 516 b Find the numbers between 1000 and 2000 that are palindromes. c The following steps change any number into a palindrome: choose any number to start with 64 reverse the digits and add 46 110 reverse the digits and add 011 repeat until you get a palindrome. 121 CHAPTER EXPLORING NUMBERS 81

Find out how many steps it takes to form a palindrome from each of these numbers. i 26 ii 28 iii 47 iv 75 v 149 vi 27 vii 1756 viii 279 ix 4021 d List some other words and place names that are palindromes. Using technology Developing number patterns Follow the instructions below to set up a spreadsheet. Enter the headings, as shown below, into the given cells in a spreadsheet. Enter 1 into cell A2 and 2 into cell A. Highlight the two cells and Fill Down (click and hold the square in the bottom right-hand corner of cell A). Fill Down to cell A1 (you should see 0 in this cell). Odd numbers 1 Enter the formula =A2 into cell B2. 2 Enter the formula =B2+2 into cell B. Click on cell B, and Fill Down to cell B1 to obtain the first 0 odd numbers. Even numbers 1 Enter the formula =A2 into cell C2. 2 Enter the formula =C2+2 into cell C. Click on cell C, and Fill Down to cell C1 to obtain the first 0 even numbers. Square numbers 1 Enter the formula =A2^2 into cell D2. 2 Click on cell D, and Fill Down to cell D1 to obtain the first 0 square numbers. Triangular numbers 1 Enter the formula =A2 into cell E2. 2 Enter the formula =E2+A into cell E. Click on cell E, and Fill Down to cell E1 to obtain the first 0 triangular numbers. 82 NEW CENTURY MATHS 7

Fibonacci numbers 1 Enter the formula =A2 into cell F2. 2 Enter the formula =F2 into cell F. Enter the formula =F2+F into cell F4. 4 Click on cell F4, and Fill Down to cell B1 to obtain the first 0 Fibonacci numbers. Questions 1 Use your spreadsheet to answer the following questions. a Name the two smallest odd numbers that are also square. Write your answer in cell H1. (Note: separate your answers with a comma.) b Name all the even numbers less than 0 that are also triangular. Write your answer in cell H2. c Find all the triangular numbers less than 60 that are also Fibonacci numbers. Write your answer in cell H. d Find all the square numbers between 00 and 600. Write your answer in cell H4. e State the 18th Fibonacci number. Write your answer in cell H5. f i In cell H6, write a formula to find the difference between the 24th and 25th Fibonacci numbers. ii What cell in column F corresponds to your answer in (i)? Write your answer in cell H7. g i Extend column A to represent the first 50 numbers. ii Extend the square and triangular numbers to represent the first 50 numbers in each pattern. h Name the first number over 1000 that is both square and triangular. Write your answer in cell H8. Extension: Factorials Factorials: a definition: 1! = 1 2! = 1 2 = 2! = 1 2 = 6... n! = 1 2 n 1 Enter the formula =A2 into cell G2. 2 Using the definition given, develop a formula for 2! in cell G, using cells G2 and A. Click on cell G4, and Fill Down to cell G1 to obtain the first 0 factorial numbers. Displaying large numbers a Highlight all the cells that don t show full numbers (e.g. 4.79E+08). Right click and choose Format Cells. Change the settings to number and 0 decimal places. b ######## in a cell indicates that the column is not wide enough to hold the number with all digits showing. You may need to widen the column until you can see all numbers down to cell G1. CHAPTER EXPLORING NUMBERS 8

Working mathematically Reasoning and communicating TLF L 199 Circus towers: square pyramids Worksheet -05 Perfect and amicable numbers Worksheet -06 Divisibility tests Figurate numbers Numbers formed from geometric shapes, such as triangular or square numbers, are called figurate numbers. There are many figurate number patterns. 1 Investigate the pentagonal numbers. 2 Investigate the hexagonal numbers. What are the names of the other types of figurate numbers? More types of numbers Investigate one or more of the following types of numbers and find out the relationships and patterns in them. You may find the Internet useful. Prepare a short talk for the class on your topic. Amicable numbers Perfect numbers The golden ratio/rectangle Irrational numbers Pythagorean triads Binary, octal and hexadecimal numbers Factorial numbers, for example the meaning of 5! -02 Tests for divisibility It is often useful to know if a number is divisible by another number. Here are some simple divisibility tests to help you. A number 2962 A number is divisible A number is is divisible by if the sum of its divisible by 4 by 2 if it digits is divisible by. if its last two ends in 0, 5 digits are 4 20 2, 4, 6 or 8. 79 is NOT divisible divisible by since 7 + 9 = 16, by 4. and does not go evenly into 16. 679 20 is divisible by 4. A number is A number is divisible 6There is no divisible by by 6 if it is divisible simple test 5 if it ends in 840 by both 2 and. for divisibility 0 or 5. ends in 8 by 7. 48 { 4 + 8 = 12 A number is A number is divisible 9 divisible by 8 if by 9 if the sum of its the last three digits is divisible by 9. A number digits are is divisible 10 divisible by 8. by 10 if it 1 592 is 74 ends in 0. divisible by 8. 8 592 9 604 26 040 82 788 100 000 000 50 005 416 5 7 279 64 805 4506 ends in 6 4 + 5 + 6 = 15 1 592 171 1 + 7 + 1 = 9 812 754 8 + 1 + 2 + 7 + 5 + 4 = 27 27 690 00 679 20 1 02004050 84 NEW CENTURY MATHS 7

Example 1 Which of the numbers 2 to 10 divide exactly into 112? Solution 2: 112 ends in a 2 so it is divisible by 2. : 1 + 1 + 2 = 4: 4 is not divisible by, so 112 is NOT divisible by. 4: 12 4 = so 112 is divisible by 4. 5: 112 does not end in 0 or 5 so is NOT divisible by 5. 6: 112 is not divisible by, so it is NOT divisible by 6. 16 7: Check by division: 7 112 so 112 is divisible by 7. 14 8: Check by division: 8 112 so 112 is divisible by 8. 9: 1 + 1 + 2 = 4: 4 is not divisible by 9 so 112 is NOT divisible by 9. 10: 112 does not end in 0 so it is NOT divisible by 10. Answer: 2, 4, 7 and 8 divide exactly into 112. Exercise -02 1 Copy this table and work out which of the numbers from 2 to 10 divide exactly into the given numbers (88 has been done for you). Ex 1 Number 2 4 5 6 7 8 9 10 252 600 88 121 6215 720 747 475 110 001 40 46 840 75 2 000 646 20 106 744 601 295 CHAPTER EXPLORING NUMBERS 85

2 Which number is divisible by both 4 and 5? Select A, B, C or D. A 10 B 15 C 20 D 25 Write a number less than 100 which is divisible by: a and 5 b 4 and 5 c 6 and 7 d 2 and 6 4 Write a number greater than 100 which is divisible by: a 6 b 5 c 7 d 2 and e 8 and 9 Just for the record Karl Gauss Karl Gauss was a German mathematician and astronomer who lived from 1777 to 1855. He invented a new way of finding the positions of heavenly bodies and was one of the first to study electricity. Gauss showed his mathematical ability early in life. When he was in primary school, the class was given the task of adding all the numbers from 1 to 100. The teacher thought this would keep the class busy for some time but Gauss was very quick to find the answer and even quicker to explain why he was not working on the problem. This is how he did it: 1 + 2 + + 4 + + 96 + 97 + 98 + 99 + 100 1 + 100 = 101 2 + 99 = 101 + 98 = 101, etc. Find how many pairs of numbers there are and then find the answer to the problem. Skillsheet -02 Factors and primes -0 Factors! The factors of a number are those whole numbers that divide exactly into it. Example 2 What are the factors of 12? Solution The possible ways of multiplying to get 12 are: 4 or 4 6 2 or 2 6 1 12 or 12 1 The factors of 12 are: 1, 2,, 4, 6 and 12. (Note that 1 will be a factor of every number.) 86 NEW CENTURY MATHS 7

Example Find the highest common factor of 24 and 0. Solution The factors of 24 are: 1, 2,, 4, 6, 8, 12, 24 The factors of 0 are: 1, 2,, 5, 6, 10, 15, 0 The common factors of 24 and 0 are: 1, 2, and 6. The highest common factor is 6. The highest common factor (HCF) of two or more numbers is the largest factor that is common to all those numbers.! Exercise -0 1 In each of these pairs, is the smaller number a factor of the larger number? a 8, 24 b, 9 c 4, 42 d 9, 45 e 8, 54 f 7, 91 g 7, 1 h 6, 48 i 5, 57 j 11, 14 2 List all the factors of: a 16 b 21 c 24 d 6 e 5 f 48 g 52 h 80 i 112 j 144 k 28 l 100 m 45 n 200 o 6 Which of these numbers is not a factor of 45? Select A, B, C or D. A 9 B 5 C 7 D 4 Find the common factors for each of these pairs of numbers. a 2, 4 b 9, 6 c 6, 14 d 8, 12 e 50, 150 f 46, 69 g 10, 15 h 12, 16 i 0, 20 j 18, 24 k 60, 90 l 9, 26 m 45, 15 n 6, 9 o 27, 64 p 50, 210 5 Find the common factors for each of these sets of numbers. a 2, 4, 6 b 10, 50, 60 c 22,, 121 d 24, 6, 144 e 6, 9, 12 f 16, 24, 40, 56 g 28, 70, 42, 98 h 0, 90, 75, 15 i 50, 60, 90, 120 6 Find the highest common factor for each of these pairs of numbers. a 12 and 60 b and 22 c 12 and 60 d 9 and 21 e 45 and 78 f 64 and 144 g 16 and 12 h 8 and 14 i 50 and 150 j 18 and 24 k 48 and 72 l 15 and 25 m 5 and 21 n 45 and 18 o 75 and 125 Ex 2 Ex CHAPTER EXPLORING NUMBERS 87

7 Every whole number has at least two factors. Is this true or false? Why? 8 Using your answers to Question 5, find the highest common factor for each of the given sets of numbers. Working mathematically Applying strategies and reasoning Factor path puzzle 1 Copy this grid, and try to reach the 100 square at the bottom corner, starting at the 200 square in the top corner and using related factors. You can move from one number to another by going sideways, up or down the page (not diagonally), but only if the numbers have a factor (not 1) in common. So you can move from 80 to 65 (common factor 5), but not from 65 to 72 (no common factor). 200 80 65 91 14 156 195 175 2 96 71 110 77 121 5 28 15 209 87 90 21 9 169 117 95 57 7 81 6 11 29 72 76 75 51 14 98 56 12 48 78 85 105 45 44 187 112 221 100 2 Find a factor path starting in the bottom corner (105) and finishing top right (195). Choose different starting and finishing positions. Do they all have connecting factor paths? Just for the record Korean mathematics In Korea, school students find the highest common factor (HCF) using the following method. To find the HCF of 24 and 0: divide by the first prime number divide by the next prime number 224 0 12 15 4 5 Since it is not possible to divide any more, stop. The HCF = 2 = 6 Use this method to find the HCF of each of the following sets of numbers. a 12 and 15 b 18 and 48 c 1, 20 and 28 d 15, 21 and 45 e 8, 12, 16 and 48 f 120 and 250 g 48 and 120 h 96 and 144 i 256, 144 and 48 j 675, 150 and 825 88 NEW CENTURY MATHS 7

Mental skills Maths without calculators Calculating differences and making change In every subtraction problem, for example 15 47, think of finding the gap between the two numbers. That is, find the number in this case that must be added to 47 to get 15. 1 Examine these examples. a 15 47 Think: 47 + = 15 47 50 5 15 Count: 47, 50, 100, 15 Add: + 50 + 5 = 88 Answer: 15 47 = 88 b 244 115 Think: 115 + = 244 50 100 150 115 5 80 44 244 Count: 115, 120, 200, 244 Add: 5 + 80 + 44 = 129 Answer: 244 115 = 129 100 150 200 c $60 $47.65 $47.65 5c $2 $10 $60 Count: $47.65, $48, $50, $60 Add: $0.5 + $2.00 + $10.00 = $12.5 Answer: $60 $47.65 = $12.5 $50 $60 $70 d $100 $88.45 $88.45 55c $1 $10 $100 Count: $88.45, $89, $90, $100 Add: $0.55 + $1.00 + $10.00 = $11.55 Answer: $100 $88.45 = $11.55 2 Now simplify the following. $80 $90 $100 a 176 88 b 221 54 c 670 56 d 425 40 e 518 89 f 199 78 g $70 $58.40 h $80 $7.25 i $45 $40.0 j $100 $69.95 k $0 $22.90 l $50 $17.10 CHAPTER EXPLORING NUMBERS 89

Working mathematically Applying strategies More factor paths Make your own 7 7 factor path grid similar to the one in the Factor path puzzle on page 88. Try making a 4 4 grid first, then a 5 5 grid and then a 7 7 grid. Discuss with your teacher the decisions you need to make as you develop your grid. Skillsheet -02 Factors and primes -04 Prime and composite numbers! A prime number has only two factors: 1 and itself. The prime numbers are 2,, 5, 7, 11, 1, 17, 19, 2,... A composite number has more than two factors. The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16,... Note: 1 is neither prime nor composite. (It has only one factor.) Exercise -04 Worksheet -07 Sieve of Eratosthenes 1 Eratosthenes, a mathematician in ancient Greece, found an easy way to work out prime numbers. It is called the Sieve of Eratosthenes and works by deleting multiples of numbers. (Use the link to go to a spreadsheet version of the Sieve.) a Copy the grid below or print out Worksheet -07. 1 2 4 5 6 7 8 9 10 11 12 1 14 15 16 17 18 19 20 21 22 2 24 25 26 27 28 29 0 1 2 4 5 6 7 8 9 40 41 42 4 44 45 46 47 48 49 50 51 52 5 54 55 56 57 58 59 60 61 62 6 64 65 66 67 68 69 70 71 72 7 74 75 76 77 78 79 80 81 82 8 84 85 86 87 88 89 90 91 92 9 94 95 96 97 98 99 100 101 102 10 104 105 106 107 108 109 110 111 112 11 114 115 116 117 118 119 120 b Cross out 1. It is neither prime nor composite. c Except for 2, colour all the multiples of 2 red. 90 NEW CENTURY MATHS 7

d Except for, colour all the multiples of green. e Continue, with different colours, until there are no more multiples. What do you notice about the numbers that are not coloured? 2 Divide these numbers into two groups (primes and composites). 10 999 2064 472 101 17 129 27 41 77 71 62 967 504 59 Write any whole numbers which are neither prime nor composite. 4 a List the prime numbers between 6 and 50. b List the composite numbers between 65 and 80. c List the prime numbers less than 20. d List the composite numbers larger than 0 but less than 47. 5 Which number is divisible only by prime numbers, itself and 1? Select A, B, C or D. A 12 B 14 C 16 D 18 6 Look up other meanings for the word composite. Suggest why this word is used the way it is in mathematics. -05 Prime factors Every composite number can be written as a product of its prime factors. The prime factors can be found by using a factor tree. Worksheet -08 Factor trees Example 4 Write 24 as a product of its prime factors. Solution Factor tree 24 Skillsheet -0 Prime factors by repeated division 8 and 8 are factors 2 4 2 and 4 are factors of 8 is prime 2 2 2 As a product of prime factors, 24 = 2 2 2. 2 is a factor of 4 2 is prime stop CHAPTER EXPLORING NUMBERS 91

Example 5 Find the highest common factor (HCF) of 1960 and 2000. Solution 1960 2000 10 196 2 1000 2 5 4 49 2 100 10 2 5 2 2 7 7 1960 = 2 2 2 5 7 7 Both numbers contain 2 2 2 5. The HCF is 2 2 2 5 = 40. 2 10 10 10 2 5 2 5 2 5 2 2000 = 2 2 2 2 5 5 5 Example 6 Write 648 as a product of its prime factors, using index notation (powers). Solution 648 So 648 = 2 2 2 = 2 4 2 24 2 4 81 4 is read to the power 4 and 4 is called the power or index. 2 2 2 9 9 2 2 2 Exercise -05 Ex 4 1 Use factor trees to express each of these numbers as a product of its prime factors. a 8 b 6 c 45 d 6 e 51 f 49 g 90 h 27 i 10 j 200 k 275 l 42 m 1250 n 1020 o 87 92 NEW CENTURY MATHS 7

2 What are the prime factors of 1260? Select A, B, C or D. A 2 5 7 B 2 2 2 5 7 C 2 2 7 D 2 2 5 7 Find the highest common factor of each of these pairs of numbers. a 24 and 486 b 6000 and 1260 c 2475 and 75 d 4900 and 1960 e 4950 and 150 f 1404 and 900 4 Use factor trees to write each number as a product of its prime factors in index notation. a 18 b 20 c 45 d 72 e 98 f 196 g 2 h 15 i 200 j 900 Ex 5 Ex 6 Just for the record Big numbers Here are the names of some very large numbers. Name Numeral Power of 10 one 1 10 0 ten 10 10 1 hundred 100 10 2 thousand 1000 10 million 1 000 000 10 6 billion 1 000 000 000 10 9 trillion 1 000 000 000 000 10 12 quadrillion 1 000 000 000 000 000 10 15 quintillion 1 000 000 000 000 000 000 10 18 sextillion 1 000 000 000 000 000 000 000 10 21 septillion 1 000 000 000 000 000 000 000 000 10 24 octillion 1 000 000 000 000 000 000 000 000 000 10 27 nonillion 1 000 000 000 000 000 000 000 000 000 000 10 0 decillion 1 000 000 000 000 000 000 000 000 000 000 000 10 Find the names of some numbers greater than a decillion. CHAPTER EXPLORING NUMBERS 9

Working mathematically Applying strategies and reasoning Goldbach s conjecture In 1742, Christian Goldbach said: Every even number greater than 2 can be written as the sum of two prime numbers. Show that his theory is true for all the even numbers between 1 and 100. Primes may be repeated, for example 10 = 5 + 5, and a number can have more than one pair of prime numbers. Skillsheet -04 Square roots and cube roots Worksheet -09 Powers and roots -06 Squares, cubes and roots Finding square roots and cubes Raising a number to the power 2 gives its square. Raising a number to the power gives its cube. Example 7 Find a the square of 11 b the cube of 5 Solution a 11 2 = 11 11 = 121 b 5 = 5 5 5 = 125 The square of 11 is 121. The cube of 5 is 125. The x 2, x and ^ keys on a calculator can be used to find the square, cube and other powers of a number. Finding square roots and cube roots The process that undoes squaring is finding the square root (symbol ). The square root of a given number is the positive value which, when multiplied by itself, produces the given number. The process that undoes cubing is finding the cube root (symbol ). Square root sign first used in 1220 Cube root sign created in 1525 94 NEW CENTURY MATHS 7

Example 8 What is: a the square root of 64? b the cube root of 27? Solution a The square root of 64 = 64 = 8 (because 8 8 = 64). b The cube root of 27 = 27 = (because = 27). Example 9 Estimate the value of 40. Solution There is no exact answer for the square root of 40, because there isn t a number which, if squared, equals 40 exactly. Instead, we find a number whose square is close to 40. Looking at the square numbers 5 2 = 25, 6 2 = 6, 7 2 = 49, we can tell 40 must be between 6 and 7. Because 40 is closer to 6 than to 49, the square root must be closer to 6. As an estimate, 40 6.. The and keys on a calculator can be used to find the square root and cube root of a number. To calculate 40, press 40 =. The result is 6.24555, a more accurate answer than our estimate above. Finding square roots and cube roots using a factor tree Example 10 Use a factor tree to find the value of 196. Solution 196 4 49 2 2 7 7 So 196 = 2 2 7 7 196 = 2 2 7 7 = 2 7 = 14 (Note: 2 2 = 2) CHAPTER EXPLORING NUMBERS 95

Example 11 Use a factor tree to find the value of 216. Solution 2 4 216 54 2 2 2 27 2 2 9 So 216 = 2 2 2 216 = 2 2 2 = 2 = 6 (Note: 2 2 2 = 2) 2 2 2 Exercise -06 Ex 7 1 Copy and complete the following table. Ex 8 Ex 8 Number 1 2 4 5 6 7 8 9 10 11 12 Number squared 16 Number cubed 512 2 Which number(s) from Question 1 are both square and cube numbers? Use your calculator to find the square of each of these numbers. a 84 b 12 c 24 d 42 4 Use your calculator to find: a 11 2 b 15 c 100 2 d 67 e 0.1 2 f.5 5 Find the square root of: a 9 b 16 c 81 d 121 e 25 f 4 g 6 h 100 6 Find the cube root of each of these numbers, using the table from Question 1. a 8 b 125 c 4 d 1000 e 729 f 27 000 7 Between which two numbers does lie? Select A, B, C or D (without using a calculator). A 4 and 5 B 5 and 6 C 26 and 28 D 756 and 784 Ex 9 27 8 Between which two numbers does 15 lie? Select A, B, C or D. A 14 and 16 B 4 and 5 C 196 and 225 D and 4 96 NEW CENTURY MATHS 7

9 Between which two consecutive whole numbers does 80 lie? 10 Give estimates for the following, then use a calculator to check. a 50 b 142 c 1000 d 66 e 999 f 12 11 Find the following square roots, using factor trees. a 484 b 1764 c 625 d 900 e 784 f 256 g 196 h 400 i 16 12 Find the following, using a factor tree. a 10 648 b 2744 c 75 d 64 000 e 491 f 9261 Ex 10 Ex 11 Using technology Absolute cell referencing Using a spreadsheet to calculate powers 1 Set up your spreadsheet by entering the information in the cells as shown below. 2 We need to use absolute cell referencing to complete this task easily. We use this technique to maintain a particular value in a cell without changing it when writing a formula. For example: to write 2 1 in cell B2, enter =$B$1^A2. This formula will not change the 2, but will change the power to each consecutive number as we Fill Down (i.e. 2 1, 2 2, 2, etc.) Click on cell B2 and Fill Down to cell B1. Your spreadsheet will now show the first 12 powers of 2. 4 By modifying the formula given in point 2, repeat this process, using the appropriate cells, absolute cell referencing and Fill Down for columns C, D and E to show the first 12 powers of, 5 and 7. CHAPTER EXPLORING NUMBERS 97

Displaying large numbers If the numbers cannot be seen properly follow these steps: a Highlight all the cells that don t show full numbers (e.g. 2 44E + 08). Right click and choose Format cells. Change the settings to number and 0 decimal places. b ######## in a cell indicates that the column is not wide enough to hold the number with all digits showing. You may need to widen the column until you can see all numbers. Using a spreadsheet to calculate the lowest common multiple 5 This activity finds the lowest common multiple (LCM) of sets of numbers. The LCM is the smallest number that all numbers in a particular set divide into; e.g. for the pair of numbers 6 and 20, the LCM = 60. This is the smallest number both 6 and 20 divide into. Find the LCM of 8, 12 and 16. Open a new sheet and enter the information shown. a In cell B2, enter the formula =$B$1*A2. Use Fill Down to find the first 15 multiples of 8. b In cells C2 and D2, enter similar formulas and Fill Down to find the first 15 multiples of 12 and 16. [Hint: Only change the absolute cell reference.] c Now, compare the columns and identify the LCM of 8, 12 and 16. 6 Modify your spreadsheet from part a above to find the LCM of the following sets of numbers. Note: you may need to extend beyond the first 15 multiples. a 6 and 15 b 12 and 18 c, 7 and 15 d 48, 60 and 75 Try other combinations of numbers and calculate each LCM. Power plus 1 Evaluate each of the following. a 7 2 b 8 4 2 c 5 2 2 d + 6 2 4 2 e 4 + + 2 + f 1 2 + 2 2 + 2 + 4 2 98 NEW CENTURY MATHS 7

2 Arrange each of these sets of index terms in order, from the smallest value to the largest. a 2, 2, 5, 5, 2 5, 5 2 b 4 4, 7, 5, 8 2, 5 2, 6 c 100 2, 1 14, 2 7, 4, 5 4 a Copy and complete: 1 2 = 11 2 = 111 2 = 1111 2 = b Based on the patterns in your part a answers, write the squares of these numbers: i 11 111 ii 1 111 111 iii 111 111 111 iv 1 111.1111 4 a Copy and complete this number pattern. 1 = 1 2 + + 4 = 1 + = (1) + (2) 5 + 6 + 7 + 8 + 9 = 8 + 27 = ( ) + ( ) 10 + 11 + 12 + 1 + 14 + 15 + 16 = + = ( ) + ( ) b Write the next two lines of the pattern in part a. c Find the sum without adding each time. Show how you did it. i 50 + 51 + + 6 + 64 ii 82 + 8 + + 99 + 100 iii 290 + 291 + + 2 + 24 iv 577 + 578 + + 624 + 625 5 Try finding the square root of each number. (They re not as hard as they look!) a 2500 b 8100 c 10 000 d 1 000 000 e 1 210 000 f 100 000 000 g 640 000 h 176 400 i 10 000 000 000 6 Find the cube root of each of these numbers. (What you discovered in Question 5 should help.) a 8000 b 4 000 c 1 000 000 d 64 000 000 e 1 000 000 000 f 27 000 7 Find the square root of: a 2 2 b 5 5 4 4 c 6 49 d 16 25 4 8 Find the cube root of: a 2 2 2 b 4 4 4 5 5 5 c 7 7 7 6 6 6 d 8 27 e 125 64 1000 f 4 729 9 Find the value of: a 9 2 b 4 c 5 4 2 6 d 2 9 e 5 6 f 4 16 10 Two prime numbers that differ by 2 are called twin primes. For example, 11 and 1 are twin primes, but 2 and 29 are not. Find the sets of twin primes between 1 and 100. CHAPTER EXPLORING NUMBERS 99

Chapter review Worksheet -10 Exploring numbers crossword Language of maths composite number cube cube root divisibility test estimate factor factor tree Fibonacci number highest common factor index notation palindrome Pascal s triangle power prime factors prime number product square square root triangular number 1 Describe in your own words how the Fibonacci numbers are formed. 2 Find the non-mathematical meaning of: a factor b index The date 0/11/0 was a palindromic date. When will be the next palindromic date? 4 Describe what a factor tree does. 5 Find as many meanings for these words as you can. a product b prime Topic overview Write in your own words what you have learnt about number patterns and the way numbers behave. What was your favourite part of this topic? What parts of this topic did you not understand? Talk to your teacher or a friend about them. Give examples of where some of the number patterns in this chapter occur or are used. This diagram provides a summary of this chapter of work. Copy it into your workbook and complete it. Use bright colours, add your own pictures, and change it, if necessary, to be sure you understand it. Triangular Factors Factor trees Composite Number patterns Fibonacci Exploring numbers Prime Divisibility tests Squares, cubes, roots 100 NEW CENTURY MATHS 7

Chapter revision Topic test 1 Write the next three numbers in each of these patterns. a 1,, 5, 7, b 2, 4, 6, 8, c 1,, 6, 10, d 1, 4, 9, 16, e 1, 1, 2,, 5, 8, f 60, 55, 50, 45, 2 In Question 1, which set of numbers are the: a square numbers? b triangular numbers? c Fibonacci numbers? Write the following. a a triangular number between 10 and 20 b the highest Fibonacci number below 40 c the next palindrome after 2002 d the next prime number after 29 e the first five composite numbers f the square number between 40 and 50 4 Write the next three lines of Pascal s triangle as shown on the right. 5 Which of the numbers from 2 to 10 divide exactly into: a 81? b 27? c 228? d 170? e 426? 6 a Write all the factors of 60. b Write all the factors of 42. 7 a Find the common factors of 42 and 60. b Find the highest common factor of 20 and 48. c Find the highest common factor of 6 and 84. 8 Find the prime numbers from: 27 9 6 29 19 9 96 1 57 2 51 100 65 7 17 1 1 67 7 8 89 27 9 Draw factor trees to find the prime factors of these numbers. a 24 b 60 c 27 d 200 e 6 f 45 g 72 h 144 10 Write your answers from Question 9 using index notation. 11 Find the square of each of the following. a 4 b 9 c 6 d 11 12 Find the square root of each of the following. a 64 b 25 c 49 d 144 1 Find the cube of each of the following. a 2 b 6 c 9 d 8 14 Find the cube root of each of the following. a 27 b 64 c 125 d 1000 15 Use factor trees to find: a 225 b 256 c 1764 d 582 16 Between which two consecutive whole numbers does lie? 1 1 1 1 2 1 1 1 1 4 6 4 1 Exercise -01 Exercise -01 Exercise -01 Exercise -01 Exercise -02 Exercise -0 Exercise -0 Exercise -04 Exercise -05 Exercise -05 Exercise -06 Exercise -06 Exercise -06 Exercise -06 Exercise -06 55 Exercise -06 CHAPTER EXPLORING NUMBERS 101

Mixed revision 1 Exercise 1-0 Exercise 1-04 Exercise 1-05 1 Use our Hindu Arabic numerals to write the Babylonian number on the right. 2 Write the Roman numeral XXIX using Hindu Arabic numerals. Use Hindu Arabic numerals to write the Chinese number shown on the right. Exercise 1-06 Exercise 1-07 Exercise 1-08 4 What is the value of the digit 6 in each of the following? a 261 b 1006 c 6 110 d 210 62 5 Write each of the following in expanded notation. a 8 b 201 c 987 6 Complete these number grids. a + 12 9 27 b top row side column 17 9 15 11 c 9 7 5 28 27 40 d top row side column 6 4 12 6 Exercise 1-09 Exercise 1-10 Exercise 1-11 Exercise 2-02 7 Find the answers to each of these. a 90 15 b 294 21 c 259 14 8 Evaluate each of the following. a 15 5 b 7 + 2 5 c 26 2 14 7 d 7 + 5 (12 ) e 414 18 f [( + 5) 2 (20 5)] 5 9 True or false? a 12 20 b 100 25 5 c 100 = 10 d 6 2 6 = 6 e 12 5 4 15 f 8 10 Measure these angles. a b c 102 NEW CENTURY MATHS 7

11 Draw an example of: a an obtuse angle b an acute angle c a reflex angle Exercise 2-04 12 Name this angle using three letters and write the name of its parts. 1 Find the complement of: b a 66 b 85 a c 12 d 89 S 14 Find the supplement of: a 2 b 90 c 105 d 15 b P A Exercise 2-01 Exercise 2-05 Exercise 2-05 15 Draw a diagram and mark in vertically opposite angles. Exercise 2-06 16 a Draw a pair of parallel lines and cut them with a transversal. b Mark a pair of alternate angles with ( ). c Mark a pair of corresponding angles with (*). d Mark a pair of co-interior angles with (+). Exercise 2-11 17 Find the size of each angle marked by a letter. a b c d Exercise 2-12 50 m 110 b p 142 a 98 e p 6 88 f a b g 8 c h m m 70 i 7 p m 56 j x 64 k x 45 75 l 58 y 122 18 Write the next three numbers in each of these patterns. a 1, 4, 7, b 1,, 6, c 1, 1, 2,, 5, d 11, 9, 7, 19 Which of the numbers from 2 to 10 divide exactly into: a 68? b 294? c 6152? 20 Find the factors of: a 18 b 45 c 60 21 Using factor trees, write each of the numbers in Question 20 as a product of its prime factors. Write your answers using index notation. 22 Simplify each of the following. a 6 2 b c 25 d 121 e 125 f 64 Exercise -01 Exercise -02 Exercise -0 Exercise -05 Exercise -06 MIXED REVISION 1 10