Working with whole numbers


 Charles Lloyd
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1 1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and negative whole numbers (integers) factors and multiples. You will learn how to: decompose integers into prime factors calculate Highest Common Factors (HCFs) and Lowest Common Multiples (LCMs) efficiently. You will also be challenged to: investigate primes. Starter: Four fours Using exactly four fours, and usual mathematical symbols, try to make each whole number from 1 to 100. Here are a few examples to start you off ( ) 5 6 You should try to stick to basic mathematical symbols such as,,, and brackets, wherever possible, but you may need to use more complicated symbols such as and! to make some of the higher numbers. Ask your teacher if you need some help with these symbols.
2 2 Chapter 1: Working with whole numbers 1.1 Addition and subtraction without a calculator You will sometimes need to carry out simple addition and subtraction problems in your head, without a calculator. These examples show you some useful shortcuts. Work out the value of When adding a string of numbers, look for combinations that add together to give a simple answer. Here, and 6 both give exact multiples of 10. Work out the value of Both these numbers are close to exact multiples of 100, so you can work out and then make a small adjustment. Work out is close to 100, so it is convenient to take away 100, then add 2 back on. Harder questions may require the use of pencil and paper methods, and you should already be familiar with these. Remember to make sure that the columns are lined up properly so that each figure takes its correct place value in the calculation.
3 1.1 Addition and subtraction without a calculator 3 Work out So Work from right to left. Add the units: Next, the 10s column: The digit 2 is entered, and the 1 is carried to the next column. Finally, the 100s column: Here are two slightly different ways of setting out a subtraction problem. You should use whichever of these methods you prefer. Work out Method For the units: 7 3 For the 10s: 2 5 cannot be done directly Exchange 10 from the 82 to give 70 and 12. Now and So Method So The first part is the same as method 1. Instead of dropping 82 down to 72, you can make 65 up to 75. Now and 8 7 1
4 Chapter 1: Working with whole numbers EXERCISE 1.1 Work out the answers to these problems in your head Use any written method to work out the answers to these problems. Show your working clearly An aircraft can carry 223 passengers when all the seats are full, but today 57 of the seats are empty. How many passengers are on the aircraft today? 20 The attendances at a theatre show were 75 (Thursday), 677 (Friday) and 723 (Saturday). How many people attended in total? 1.2 Multiplication without a calculator You will sometimes need to carry out simple multiplication problems in your head. This example shows one useful shortcut. Work out the value of is almost 50, so you can work out 50 3 then take off the extra So Harder questions will require pencil and paper methods. Here is a reminder of how short multiplication works. Work out the value of
5 1.2 Multiplication without a calculator So Begin with Enter as 8 with the 1 carried. Next, 7 6 2, plus the 1 carried, makes 3. Enter as 3 with the carried. Finally, , plus the carried, makes 16. Entered as 6 with the 1 carried; enter this 1 directly into the 1000s column. When working with bigger numbers, you will need to use long multiplication. There are two good ways of setting this out use whichever one you are most confident with. Work out the value of Method First, multiply 92 by to give Next, prepare to multiply 92 by 30, by writing a zero in the units column. This guarantees that you are multiplying by 30, not just times 3 gives 176. Finally, add 1968 and to give the answer So
6 6 Chapter 1: Working with whole numbers Method First, write 92 and 3 along the top and down the end of a rectangular grid Next, add diagonal lines, as shown Within each square of the grid, carry out a simple multiplication as shown. For example, 9 times 3 is Finally, add up the totals along each diagonal, starting at the right and working leftwards. EXERCISE 1.2 Use short multiplication to work out the answers to these calculations Use any written method to work out the answers to these problems. Show your working clearly
7 1.3 Division without a calculator 7 17 A company has 23 coaches and each coach can carry 55 passengers. What is the total number of passengers that the coaches can carry? 18 I have a set of 12 encyclopaedias. Each one has 199 pages. How many pages are there in the whole set? 19 Joni buys 16 stamps at 19 pence each and 13 stamps at 26 pence each. How much does she spend in total? 20 A small camera phone has a rectangular chip of pixels that collect and form the image. The chip size is 320 pixels long and 20 pixels across. Calculate the total number of pixels on the chip. 1.3 Division without a calculator Division is usually more awkward than multiplication, but this example shows a helpful method if the number you are dividing into (the dividend) is close to a convenient multiple of the number you are dividing by (the divisor). Work out the value of is So is almost 700, so you can work out then take off the extra 7 7 In most division questions you will need to use a formal written method. Here is an example of short division, with a remainder. Work out the value of remainder 1 So r 1 (or ) First, set the problem up using this division bracket notation. Divide into 6: it goes 1 time, with a remainder of 2. Next, divide into 27: it goes 6 times, with a remainder of 3. Finally, divide into 33: it goes 8 times, with a remainder of 1.
8 8 Chapter 1: Working with whole numbers When dividing by a number bigger than 10, it is usually easier to set the working out as a long division instead. The next example reminds you how this is done. Work out the value of Begin by setting up the problem using division bracket notation. 13 will not divide into 3, so divide 13 into 33. This goes 2 times, with remainder Bring down the next digit, 0 in this case, to make the 7 up to divides into 70 five times, with remainder 5. Finally, bring down the digit 2 to make divides into 52 exactly times, with no remainder. So exactly
9 1. Positive and negative integers 9 EXERCISE 1.3 Use short division to work out the answers to these calculations. (Four of them should leave remainders.) Use long division to work out the answers to these problems. Show your working clearly. (Only the last two should leave remainders.) grams of chocolate is shared out equally between 6 people. How much does each one receive? 18 In a lottery draw the prize of 3250 is shared equally between 13 winners. How much does each receive? 19 Seven children share 100 sweets in as fair a way as possible. How many sweets does each child receive? 20 On a school trip there are 16 teachers and 180 children. The teachers divide the children up into equalsized groups, as nearly as is possible, with one group per teacher. How many children are in each group? 1. Positive and negative integers It is often convenient to visualise positive and negative whole numbers, or integers, placed in order along a number line. The positive integers run to the right of zero, and negative integers to the left: Smaller at this end larger at this end. Mathematicians describe numbers on the right of the number line as being larger than the numbers on the left. This makes sense for positive numbers, where 6 is obviously bigger than, for example, but care must be taken with negative numbers. is bigger than 6, for example, and 8 is smaller than 7. You need to be able to carry out basic arithmetic using positive and negative numbers, with and without a calculator. Many calculators carry two types of minus sign key: one for marking a number as negative, and another for the process of subtraction. So, in a calculation such as 6 5, you have to start with the quantity 6 and then subtract 5. Subtraction means moving to the left on the number line, so the answer is
10 10 Chapter 1: Working with whole numbers Take care when two minus signs are involved: the rule that two minuses make a plus is not always trustworthy. For example, (two minuses make even more minus!), whereas So two adjacent minus signs are equivalent to a single plus sign. If two adjacent signs are the same: or then the overall sign is positive. And if the signs are different: or then the overall sign is negative. Without using a calculator, work out the values of: a) 6 9 b) 5 c) 8 3 d) 5 6 a) b) 5 1 c) d) When multiplying or dividing with positive or negative numbers, it is usually simplest to ignore the minus signs while you work out the numerical value of the answer. Then restore the sign at the end. If an odd number of negative numbers is multiplied or divided, the answer will be negative. If an even number of minus signs is involved, the answer will be positive. Without using a calculator, work out the values of: a) ( 5) () b) ( ) ( 3) c) ( 8) ( 2) d) 5 ( ) ( 2) a) ( 5) () 20 b) ( ) ( 3) 12 c) ( 8) ( 2) d) 5 ( ) ( 2) and there are two minus signs, so the answer is positive. EXERCISE 1. Without using a calculator, work out the answers to the following: 1 ( 6) 2 6 ( 3) 3 3 ( 2) 2 ( 1) ( 5) (5) 5
11 1.5 Factors, multiples and primes Arrange these in order of size, smallest first: 8, 3, 5, 1, Arrange these in order of size, largest first: 12, 13, 5, 9,. 23 What number lies midway between and 12? 2 What number lies onethird of the way from 10 to 2? 1.5 Factors, multiples and primes You will remember these definitions from earlier work: A multiple of a number is the result of multiplying it by a whole number. The multiples of are, 8, 12, 16, A factor of a number is a whole number that divides exactly into it, with no remainder. The factors of 12 are 1, 2, 3,, 6, 12. A prime number is a whole number with exactly two factors, namely 1 and itself. The number 1 is not normally considered to be prime, so the prime numbers are 2, 3, 5, 7, 11, If a large number is not prime, it can be written as the product of a set of prime factors in a unique way. For example, 12 can be written as A factor tree is a good way of breaking a large number into its prime factors. The next example shows how this is done. Write the number 180 as a product of its prime factors Begin by splitting the 180 into a product of two parts. You could use 2 times 90, or times 5, or 9 times 20, for example. The result at the end will be the same in any case. Here we begin by using 18 times 10. Since neither 18 nor 10 is a prime number, repeat the factorising process.
12 12 Chapter 1: Working with whole numbers has been broken down into 9 times 2, and 10 into 2 times 5. The 2s and the 5 are prime, so they are circled and the tree stops there. The 9 is not prime, so the process can continue The factor tree stops growing when all the branches end in circled prime numbers. 3 3 Thus means the factor 2 is used twice (two squared). If it had been used three times, you would write 2 3 (two cubed). EXERCISE List all the prime numbers from 1 to 0 inclusive. You should find that there are 12 such prime numbers altogether. 2 Use your result from question 1 to help answer these questions: a) How many primes are there between 20 and 0 inclusive? b) What is the next prime number above 31? c) Find two prime numbers that multiply together to make 03. d) Write 91 as a product of two prime factors. 3 Use the factor tree method to obtain the prime factorisation of: a) 80 b) 90 c) 50 Use the factor tree method to obtain the prime factorisation of: a) 36 b) 81 c) 1 What do you notice about all three of your answers? 5 When 56 is written as a product of primes, the result is 2 a b where a and b are positive integers. Find the values of a and b. 1.6 Highest common factor, HCF Consider the numbers 12 and 20. The number 2 is a factor of 12, and 2 is also a factor of 20. Thus 2 is said to be a common factor of 12 and 20. Likewise, the number is also a factor of both 12 and 20, so is also a common factor of 12 and 20.
13 1.6 Highest common factor, HCF 13 It turns out that 12 and 20 have no common factor larger than this, so is said to be the highest common factor (HCF) of 12 and 20. You can check that really is the highest common factor by writing 12 as 3 and 20 as 5; the 3 and 5 share no further factors. Find the highest common factor (HCF) of 30 and 80. By inspection, it looks as if the highest common factor may well be 10. Check: , and and clearly 3 and 8 have no further factors in common. So HCF of 30 and 80 is 10 By inspection means that you can just spot the answer by eye, without any formal working. There is an alternative, more formal, method for finding highest common factors. It requires the use of prime factorisation. Use prime factorisation to find the highest common factor of 30 and 80. By the factor tree method: Similarly, So HCF of 30 and Look at the 2 s: 30 has one of them, 80 has four. Pick the lower number: one 2 Look at the 3 s: 30 has one of them, but 80 has none. Pick the lower number: no 3s Look at the 5 s: 30 has one of them, and 80 has one. Pick the lower number: one 5 The prime factorisation method involves a lot of steps, but it is particularly effective when working with larger numbers, as in this next example. Use prime factorisation to find the highest common factor of 96 and 156. By the factor tree method: and HCF of 96 and
14 1 Chapter 1: Working with whole numbers It is important to be able to use the prime factorisation method in case it appears as an IGCSE examination question. You might like to try this ingenious alternative approach. A Greek mathematician named Euclid used it 3500 years ago, so it is often known as Euclid s method. Use Euclid s method to find the HCF of 96 and 156. [96, 156] [60, 96] [36, 60] [2, 36] [12, 2] [12, 12] Begin by writing the two numbers in a square bracket. Each new bracket contains the smaller of the two numbers, and their difference. Stop when both numbers are equal. So HCF (96, 156) 12 EXERCISE Use the method of inspection to write down the highest common factor of each pair of numbers. Check your result in each case. a) 12 and 18 b) 5 and 60 c) 22 and 33 d) 27 and 5 e) 8 and 27 f) 26 and Write each of the following numbers as the product of prime factors. Hence find the highest common factor of each pair of numbers. a) 20 and 32 b) 36 and 60 c) 80 and 180 d) 72 and 108 e) 120 and 195 f) 1 and Use Euclid s method to find the highest common factor of each pair of numbers. a) 12 and 30 b) 2 and 36 c) 96 and 120 d) 90 and 10 e) 78 and 102 f) 8 and Lowest common multiple (LCM) Consider the numbers 15 and 20. The multiples of 15 are 15, 30, 5, 60, 75, The multiples of 20 are 20, 0, 60, 80, Any multiple that occurs in both lists is called a common multiple. The smallest of these is the lowest common multiple (LCM). In this example, the LCM is 60. There are several methods for finding lowest common multiples. As with highest common factors, one of these methods is based on prime factorisation.
15 1.7 Lowest common multiple (LCM) 15 Find the lowest common multiple of 8 and 180. First, find the prime factors of each number using a factor tree if necessary Look at the powers of 2: Next, the powers of 3: Finally, the powers of 5: There are factors of 2 in 8, but only 2 in 180. Pick the higher of these: There is 1 factor of 3 in 8, but 2 in 180. Pick the higher of these: 2 There is no factor of 5 in 8, but 1 in 180. Pick the higher of these: 1 Putting all of this together: LCM of 8 and An alternative method is based on the fact that the product of the LCM and the HCF is the same as the product of the two original numbers. This gives the following result: a b LCM of a and b HCF of a and b This can be quite a quick method if the HCF is easy to spot. Find the lowest common multiple of 70 and 110. By inspection, HCF is 10 So: LCM
16 16 Chapter 1: Working with whole numbers It is also possible to find the HCF and LCM of three (or more) numbers. The prime factorisation method remains valid here, but other shortcut methods can fail. This example shows you how to adapt the factorisation method when there are three numbers. Find the HCF and LCM of 16, 2 and 28. Write these as products of prime factors: HCF of 16, 2 and 28 is 2 2 LCM of 16, 2 and 28 is The lowest number of 2s from 2 or 2 3 or 2 2 is 2 2 The highest number of 2s from 2 or 2 3 or 2 2 is 2 EXERCISE 1.7 Find the lowest common multiple (LCM) of each of these pairs of numbers. You may use whichever method you prefer and and and 5 25 and and and and and and and and and and and and and a) Write 60 and 8 as products of their prime factors. b) Hence find the LCM of 60 and a) Write 66 and 99 as products of their prime factors. b) Hence find the LCM of 66 and 99. c) Find also the HCF of 66 and a) Write 10, 36 and 56 as products of their prime factors. b) Work out the Highest Common Factor, HCF, of 10, 36 and 56. c) Work out the Lowest Common Multiple, LCM, of 10, 36 and a) Write 0, 8 and 600 as products of their prime factors. b) Work out the Highest Common Factor, HCF, of 0, 8 and 600. c) Work out the Lowest Common Multiple, LCM, of 0, 8 and 600.
17 Review exercise Virginia has two friends who regularly go round to her house to play. Joan goes round once every days and India goes round once every 5 days. How often are both friends at Virginia s house together? 22 Eddie owns three motorcycles. He cleans the Harley once every 8 days, the Honda once every 10 days and the Kawasaki once every 15 days. Today he cleaned all three motorcycles. When will he next clean all three motorcycles on the same day? REVIEW EXERCISE 1 Work out the answers to these arithmetic problems, using mental methods. Written working not allowed! Use pencil and paper methods (not a calculator) to work out the answers to these arithmetic problems Work out the answers to these problems using negative numbers. Do not use a calculator. 21 ( 7) ( 1) 22 6 ( 3) 23 ( 10) ( 13) ( 8) ( 3) 3 31 Use a factor tree to find the prime factorisation of: a) 70 b) 12 c) 96 d) a) Find the Highest Common Factor (HCF) of 2 and 56. b) Find the Lowest Common Multiple (LCM) of 2 and a) Write down the Highest Common Factor (HCF) of 20 and 22. b) Hence find the Lowest Common Multiple (LCM) of 20 and a) Write 360 in the form 2 a 3 b 5 c b) Write as an ordinary number.
18 18 Chapter 1: Working with whole numbers 35 Who is right? Explain carefully. If the HCF of two numbers is 1, then they must both be primes. Not necessarily true. Chuck Lilian 36 Pens cost 25p each. Mr Smith spends 120 on pens. Work out the number of pens he gets for The number 110 can be written as 3 2 c d, where c is a whole number and d is a prime number. Work out the value of c and the value of d. [Edexcel] [Edexcel] 38 a) Express 72 and 96 as products of their prime factors. b) Use your answer to part a) to work out the Highest Common Factor of 72 and 96. [Edexcel] Key points 1 Mental methods can be used for simple arithmetic problems. When adding up strings of whole numbers, look for combinations that add up to multiples of Harder addition and subtraction problems require formal pencil and paper methods. Make sure that you know how to perform these accurately. 3 Simple multiplication problems may be done mentally or by short multiplication. For harder problems, you need to be able to perform long multiplication reliably. If you find the traditional columns method awkward, consider using the box method instead both methods are acceptable to the IGCSE examiner. Long division is probably the hardest arithmetic process you will need to master. The traditional columns method is probably the best method there are alternatives, but they can be clumsy to use. If you have a long division by 23, say, then it may be helpful to write out the multiples 23, 6, 69,, 230 before you start. 5 Exam questions may require you to manipulate and order negative numbers. Remember to treat the two minuses make a plus rule with care, for example, 2 3 6, but Nonprime whole numbers may be written as a product of primes, using the factor tree method. This leads to a powerful method of working out the Highest Common Factor or Lowest Common Multiple of two numbers. 7 Sometimes you may be able to spot HCFs or LCMs by inspection. This result might help you to check them: a b LCM of a and b HCF of a and b
19 Internet Challenge 1 19 Internet Challenge 1 Prime time Here are some questions about prime numbers. You may use the internet to help you research some of the answers. 1 Find a list of all the prime numbers between 1 and 100, and print it out. How many prime numbers are there between 1 and 100? 2 Find a list of all the prime numbers between 1 and How many prime numbers are there between 1 and 1000? Compare your answers to questions 1 and 2. Does it appear that prime numbers occur less often as you go up to larger numbers? 3 Why is 1 not normally considered to be prime? How many British Prime Ministers had held office, up to the resignation of Gordon Brown in 2010? Is this a prime number? 5 Is there an infinite number of prime numbers? 6 What is the largest known prime number? 7 Is there a formula for finding prime numbers? 8 Where is the Prime Meridian? 9 Find out how the Sieve of Eratosthenes works, and use it to make your own list of all the primes up to 100. Check your list by comparing it with the list you found in question Some primes occur in adjacent pairs, which are consecutive odd integers, for example, 11 and 13, or 29 and 31. Find some higher examples of adjacent prime pairs. How many such pairs are there?
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