Number. 1.1 Properties of whole numbers CHAPTER. Example 1. Example 2. Example 3

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1 Number 1 CHAPTER 1.1 Properties of whole numbers A factor of a number, x, is a number which divides into x an exact number of times. So 3 is a factor of 1 because and are called a factor pair of 1 3 is not a factor of 10 because 10 leaves a remainder when divided by 3 A multiple of a number, y, is a number which divides exactly by y. The first three multiples of 5 are 5, 10 and 15 1 is not a multiple of 5 because 1 leaves a remainder when divided by 5 Example 1 Find all the factors of 0 Solution 1 1, 0, 10, 5 Write them in factor pairs. A number which has exactly two factors is called a prime number. So,, 3, 5 and 3 are prime numbers but 1, and 15 are not. 1 is not a prime number because it only has one factor, 1 Example is the only even prime number. Explain why. Solution is a prime number because it has exactly two factors, 1 and All other even numbers have at least 3 factors, 1, and the number itself. A common factor of two numbers, x and y, is a number which is both a factor of x and is also a factor of y. So is a common factor of 1 and 0 3 is not a common factor of 1 and 0 because 3 is not a factor of 0 Example 3 In each case the product of the two numbers is 0 Find all the common factors of 1 and 0 Solution 3 1,,3,,6,1 1,,,5,10, 0 1,, are the common factors of 1 and 0 These are the factors of 1 These are the factors of 0 These three factors appear in both lists. 1

2 CHAPTER 1 Number Exercise 1A 1 Which of the following numbers are factors of 18? a 1 b 6 c 9 d 3 e 36 Which of the following numbers are factors of 30? a 1 b 0 c 15 d 3 e 6 3 Find all the factors of 50 Write them in factor pairs. List all of the factors of the following numbers. a 8 b 10 c 16 d e 8 f 3 g 36 h 0 i 60 j List all the common factors of a 6 and 8 b 6 and 9 c 6 and 10 d 8 and 1 e 1 and 15 f 10 and 0 g 15 and 0 h 18 and 6 a Write down the first three multiples of b Write down the first three multiples of 10 c Write down the first four multiples of 8 d Write down the first four multiples of 7 e Write down the first three multiples of 3 7 State whether the following statements are true or false. a 1 is a multiple of b 1 is a factor of 7 c is a multiple of 3 d 7 is a multiple of 9 e 1 is both a multiple of 6 and a factor of 36 f 9 is a factor of 7 g 6 is a multiple of 1 h is a multiple of 1 8 Show that 33 is a factor of Find the first multiple of 9 which is greater than Find two prime numbers between 110 and Bertrand s theorem states that Between any two numbers n and n, there always lies at least one prime number, providing n is bigger than 1. Show that Bertrand s theorem is true i for n 10 ii for n 3 1 Find a number which has exactly a factors b 3 factors c 7 factors d 10 factors 1. Multiplication and division of directed numbers A directed number is a number with a or a sign. and 3 are examples of directed numbers. Often a directed number is written in brackets, for example () and (3). Multiplication (3) is the same as 3 () is the same as so (3) () is the same as also means 3 () means () () () so 3 () or (3) () (6)

3 1. Multiplication and division of directed numbers CHAPTER 1 Look at the patterns in the multiplications on the right. The blue numbers are decreasing by 1 The orange numbers are increasing by The pattern continues like this. The rules are () () () positive positive positive () () () positive negative negative () () () negative positive negative () () () negative negative positive (3) () (6) () () () (1) () () 0 () 0 (1) () () () () () (3) () (6) Division and 6 3 (3) () (6), so (6) (3) () and (6) () (3) The rules are () () () positive positive positive () () () positive negative negative () () () negative positive negative () () () negative negative positive When multiplying or dividing two directed numbers you can remember the rules by the following if the signs are the same, the answer is positive if the signs are different, the answer is negative. Example a Work out (5) (3) Solution a (5) (3) (15) (15) is also written as 15 b (16) () (8) (8) is also written as 8 b Work out (16) () The signs are different so the answer is negative and The signs are the same so the answer is positive and 16 8 Exercise 1B 1 Work out a () () b (3) (5) c () (6) d (3) (5) e () (5) f () (5) g (3) (8) h (1) (9) i () () Work out a (6) (3) b (8) () c (10) (5) d (1) (3) e (8) () f (1) (1) g (1) () h (1) () 3 Find the missing directed number. a (10) ( ) () b (8) ( ) () c (3) ( ) (1) d (5) ( ) (0) e (5) ( ) (5) f ( ) () (0) g ( ) (3) () h ( ) () (5) i (16) ( ) () 3

4 CHAPTER 1 Number Work out the product of a (6) and (3) b (5) and ( ) c (3) and (5) d (6) and (6) e (3) and () f () and (9) g (5) and () h (3) and () 1.3 Squares and cubes The square of a number, x, is the number which is the product x x. The square of the number x is written x. So the square of 10 is written as The square of 10 is 100 The cube of a number, y, is the number which is the product y y y. The cube of the number y is written y 3. So the cube of is written 3 The cube of is 6 The square root of a number n is the number which when squared gives n. The square root of the number n is written n. So the square root of 16, written 16,is since 16 Since () 16, the negative square root of 16 is It is not possible to find the square root of a negative number. The cube root of a number m is the number which when cubed gives m. The cube root of a number m is written 3 m. So the cube root of 1000, written 3, 1000 is 10 since It is possible to find the cube root of a negative number. Example 5 Work out Solution because (3) (3) (3) 7 8 (3) 5 Exercise 1C 1 Work out a 3 b 5 c 11 d 13 e 15 f 100 Work out a () b () c (10) d (1) 3 Work out a 3 3 b 1 3 c 5 3 d (10) 3 e () 3 Work out a 3 b c 3 3 d 3 8 e f 3 3 g (1) 3 3 (3) 3 h (3) 3 i j 6

5 1. Index laws CHAPTER 1 5 Here is a number pattern (1 ) (1 3) Show that the next line of the number pattern is also true. 1. Index laws As well as squares and cubes it is possible to represent a number multiplied by itself any number of times. For example, ( raised to the power ) means 3 6 (3 raised to the power 6) means Another name for power is index. Example 6 Work out a 3 b 6 Solution 6 a b 6 6 To work out one number raised to a power multiplied by the same number raised to a second power you add the powers. For example 3 7 because 3 and and ( ) ( ) 3 7 To divide one number raised to a power by the same number raised to a second power you subtract the powers For example, cancelling all the 3s on the bottom with four of the 3s on the top. So Example 7 a Work out 5. Give your answer as a power of b Work out Give your answer as a power of 5 c Work out (3 ).Give your answer as a power of 3 d Work out 7. Give your answer as a power of Solution 7 a b c (3 ) d

6 CHAPTER 1 Number Example Work out Solution Work out means evaluate the expression rather than leaving the answer as a power of 7 Exercise 1D 1 Write as a power of a 5 b 3 c 6 d 3 e 6 Write as a power of 3 a 3 3 b c 3 3 d e Write as a power of a single number a b c 3 3 d e Find the value of n a 3 n b n 8 c 5 n 10 d 3 n e 6 3 n 5 Work out a 3 3 b 5 3 c 5 d e Write as a power of 3 a b (3 3 ) c d e Write as a power of a single number a 3 5 b c d e Work out a b c d e Work out the value of n in the following. a 0 5 n b 3 n c 50 5 n d 8 3 n e 5 3 n Order of operations Some expressions include powers and other operations. BIDMAS gives the order in which operations should be carried out. Remember that BIDMAS stands for Brackets Indices Division Multiplication Addition Subtraction

7 1.5 Order of operations CHAPTER 1 If there are brackets, work out the value of the expression in the brackets first. Square roots are carried out at the same stage as indices. If there are no brackets, do multiplication and division before addition and subtraction no matter where they come in an expression. If an expression has only addition and subtraction then work it out from left to right. Example 9 Work out a 6 b c 16 Solution 9 a b Squaring is carried out before multiplication. Squaring and cubing are carried out before addition. c Working out indices is carried out before division. Example 10 Work out a ( ) b (1 5) 3 c 9 Solution 10 a 8, then 8 6 Expressions in brackets are worked out first, then indices. b (1 5) 3 (1 10) c 3 1 Multiplication is carried out before subtraction. Square root is found before multiplication is carried out. Exercise 1E 1 Work out a 5 b c 3 d 5 10 e 3 f 6 g 8 1 h 5 Work out a ( 5) b (7 3) c (5 5) d (1 5) e ( ) f (8 5) g (0 10) h (18 9) 3 Work out a 0 3 b 17 c 17 d 7 e 36 f 5 5 g 1 h 5 Work out a b c 50 5 d e 36 3 f (10 ) g (1 ) h (15 5) i (1 3) 5 Work out a 3 b 3 c 5 d 6 3 e 10 3 f 6 1 g 6 h 7 3 i 6 8 7

8 CHAPTER 1 Number 6 Work out a 9 b c d e 9 5 f 5 6 g 36 5 h Using a calculator Arithmetical expressions can be worked out using a scientific calculator. To work out 6. key in 6. x which gives 0.96 For working out cubes, some scientific calculators have a cube key. To work out key in 1. 1 x 3 which gives All scientific calculators have a power or index key. This comes in two forms. The first form is a y x key. To work out key in 1. 1 y x 3 which gives The second form is an upwards arrow key. To work out key in which gives Not all calculators are the same so make sure you know how to calculate powers on your own calculator. x 3 Example 11 Use a calculator to work out a.5 b Solution 11 a Key in. 5 x b Key in. 3 y x 3 or key in. 3 3 Write down the answer Key in 3. 1 y x or key in 3. 1 Write down the answer Finally add the two results. A more efficient method would be to key in. 3 y x y x which does not involve writing down the separate results for.3 3 and 3.1. Sometimes an answer has to be given correct to one decimal place. The answer to b correct to one decimal place is 10.5 The usual method of finding a square root is to use the square root key on the calculator. All scientific calculators have a square root key. 8

9 1.6 Using a calculator CHAPTER 1 Example 1 Work out Solution 1 The numerator is 0. The denominator is 6 The answer is 3.0 Key in. y x Key in 3. 5 x. 5 x Finally divide 0. by 6 The expression can be worked out more efficiently using the key sequence (. y x ) ( 3. 5 x. 5 x ) 1 The reciprocal of a number n is the number or 1 n. n The reciprocal of is 1 or 0.5 The reciprocal of 1.5 is 0.8 When a number is multiplied by its reciprocal the answer is always 1 All numbers, except 0, have a reciprocal. The reciprocal button on a calculator is usually shown as 1 1 or x 1. x Dividing an expression by a number is the same as multiplying the expression by the reciprocal of that number. Exercise 1F 1 Work out a 8. b 9. c (3.6) d e 15. Work out a 13 1 b c 37 3 d Work out a 3 31 b 5 3 c 19 d 35 5 Work out a 00 1 b 0 1. c d Work out a 576 b 10 c 65 d Work out, giving your answers correct to one decimal place a 00 b 300 c 80 d 18 e 15 7 Work out, giving your answers correct to one decimal place a. 3 b 3.7 c d (1.7) 6 e Work out, giving your answers correct to one decimal place a b c d Work out, giving your answers correct to one decimal place a b ( ) 3 c 3. (.7) d

10 CHAPTER 1 Number 10 Work out, giving your answers correct to one decimal place a b c d e f g h i j Find the reciprocal of each of the following numbers. a b 8 c 0 d 0.65 e Prime factors, HCF and LCM A prime factor of the number n, is a prime number which is a factor of n. The factors of 30 are 1,, 3, 5, 6, 10, 15 and 30 The prime numbers in this list are, 3 and 5 So the prime factors of 30 are, 3 and 5 Prime numbers can be thought of as the basis of all whole numbers because all whole numbers are either prime or can be written as a product of prime numbers. For example, 15 is not prime, but can be written as the product is not prime, but can be written as the product 3 For small numbers it is easy to see what prime numbers to use. For larger numbers use the following method. Example 13 Write 7 as a the product of its prime factors Solution 13 a The prime factors of 7 are and 3 7 Divide 7 by b the product of powers of its prime factors Divide 36 by Divide 18 by Divide 9 by Divide 3 by b The highest common factor (HCF) of two numbers is the largest number which is a factor of both of the numbers. For example, the highest common factor (HCF) of 8 and 1 is because it is the largest number that is a factor of both 8 and 1 For larger numbers, it is useful to list the factors of each number and then pick out the largest number that appears in all the lists.

11 1.7 Prime factors, HCF and LCM CHAPTER 1 Example 1 Find the highest common factor (HCF) of and 36 Solution 1 The factors of are 1,, 3,, 6,8,1, The factors of 36 are 1,, 3,, 6, 9, 1, 18, 36 The numbers which appear in both lists, that is the common factors, are 1,, 3,, 6 and 1 So 1 is the highest common factor of and 36 The lowest common multiple (LCM) of two numbers is the smallest number which is a multiple of both numbers. For example, the lowest common multiple of 8 and 1 is because it is the smallest number which is a multiple of both 8 and 1 For larger numbers, it is useful to list the multiples of each number and then pick out the smallest number that appears in both lists. Example 15 Find the lowest common multiple (LCM) of 15 and 0 Solution 15 The first few multiples of 15 are 15, 30, 5, 60, 75, The first few multiples of 0 are 0, 0, 60,80, 100, The smallest number which appears in both lists is 60 So the lowest common multiple of 15 and 0 is 60 The HCF can be worked out for large numbers if each of the numbers is written as a product of its prime factors. For example, for the numbers 10 and 1 the products are So 3 is the highest common factor of 10 and 1 In terms of products of powers of their prime factors, Their highest common factor ( 3 3) is the product of the lowest power of each of their common prime factors. Example 16 Find the highest common factor (HCF) of 750 and 5 Solution HCF Write 750 and 5 as the product of powers of their prime factors. The common prime factors are 3 and 5 The lowest power of 3 is 1 (as ) and the lowest power of 5 is 11

12 CHAPTER 1 Number To find the LCM of 36 and 10, list the multiples of 36 and 10 until the same multiple appears in both lists. The multiples of 36 are 36, 7, 108, 1, 180, 16, 5, 88, 3, 360, The multiples of 10 are 10, 0, 360, The LCM of 36 and 10 is 360 As a product of its prime factors, As a product of their prime factors the numbers 36 and 10 are The LCM of 36 and 10 (360) is the product of the common prime factors and all other prime factors, that is In terms of products of powers of their prime factors, Their lowest common multiple, 360 ( 3 3 5), is the product of the highest power of all their prime factors. Example 17 Find the lowest common multiple (LCM) of 750 and 5 Solution LCM Write 750 and 5 as the product of powers of their prime factors. The highest power of is 1 (as 1 ) The highest power of 3 is The highest power of 5 is 3 Exercise 1G 1 Find the two prime numbers that are between 30 and 0 Find two prime numbers which have a sum of 7 3 Find two prime numbers which have a product of 1 Find two prime numbers which are factors of 0 5 Find two prime numbers which are factors of 6 Find two prime numbers which are factors of 33 7 Write the following numbers as a product of two prime factors. a 10 b 15 c 1 d e 33 f 39 8 Which of the following show a number written correctly as a product of prime factors? a 1 3 b 18 9 c 0 5 d 16 e 56,,, 7 f Write each of these numbers as a product of its prime factors. a 30 b c 8 d 36 e 60 f 63 g 5 h 80 i 76 j 88 k 68 l 66 1

13 1.7 Prime factors, HCF and LCM CHAPTER 1 10 Find the highest common factor (HCF) of the following pairs of numbers. a 1 and 1 b 6 and 9 c 6 and 8 d 8 and 10 e 6 and Find the highest common factor (HCF) of the following pairs of numbers. a 1 and 18 b 10 and 15 c 16 and 0 d 18 and e and 30 1 Find the lowest common multiple (LCM) of the following pairs of numbers. a 6 and 8 b 6 and 9 c 6 and 10 d 9 and 1 e 10 and Find the lowest common multiple (LCM) of the following pairs of numbers. a 1 and 15 b 1 and c 1 and 18 d 18 and e 0 and 1 a Find the number of multiples of 3 that are less than 100 b Find the number of multiples of 5 that are less than Frank has two flashing lamps. The first lamp flashes every seconds. The second lamp flashes every 6 seconds. Both lamps start flashing together. a After how many seconds will they again flash together? b How many times in a minute will they flash together? 16 As a product of its prime factors, Write 70 as a product of its prime factors. 17 The number 8 can be written in the form n 3 Find the value of n. 18 The number 189 can be written in the form 3 n p where n and p are prime numbers. Find the value of n and the value of p. 19 The number 10 can be written in the form n m p where n, m and p are prime numbers. Find the value of each of n, m and p. 0 x 3 5, y a Find the highest common factor (HCF) of x and y. b Find the lowest common multiple (LCM) of x and y a Which of the numbers in the list are factors of 88? b Which of the numbers in the list are factors of 550? Write each of these numbers as a product of its prime factors. a 105 b 539 c 31 d 87 e Find the lowest common multiple of these pairs of numbers. a and 30 b 7 and 36 c 8 and 35 d 36 and e 5 and 7 Find all the integer values of n less than or equal to 10 for which n 1 is a prime number. 5 a Any square number which is even is always a multiple of. Explain why. b Investigate what the corresponding answer is for square numbers which are odd. c Explain why the number cannot be a square number. 13

14 CHAPTER 1 Number Chapter summary You should now know that: the power or index of a number shows how many of the number are multiplied together, for example, in 5 the 5 is the power or index and five s are multiplied together, that is the square root of a number is that number which when squared gives the original number the negative square root of a number is that negative number which when squared gives the original number the correct order of working out an expression is obtained by using BIDMAS the reciprocal of a whole number n is the fraction n 1 the highest common factor (HCF) of two numbers is the largest number which is a factor of both of the numbers the lowest common multiple (LCM) of two numbers is the smallest number which is a multiple of both numbers. You should also be able to: use the rule for adding powers when two of the same number raised to a power are multiplied together use the rule for subtracting powers when one number raised to a power is divided by the same number raised to a power use a calculator to work out the square root of a number use a calculator to work out powers of a number write any whole number as a product of its prime factors find the highest common factor of two or more numbers find the lowest common multiple of two or more numbers. Chapter 1 review questions 1 Work out a 0 b 7000 c 100 d 900 Work out a 3 b 10 3 c 5 3 d 1 3 e Work out a 5 b 3 3 c d Work out a 3 5 b 36 c 10 9 d Work out a 8 b 3 8 c d

15 Chapter 1 review questions CHAPTER 1 6 Work out a 5 b 10 c (9) d (5 1) e ( 3) 7 Write each of the following as a single power of a 3 b 6 c ( 3 ) d 8 e 6 8 Write each of the following as a single power of 3 a 3 3 b c Each of the following represents a number written as a product of powers of its prime factors. Find the number. a 3 3 b c a Express 108 as a product of powers of its prime factors. b Find the highest common factor (HCF) of 108 and (1387 June 00) 11 a Express 10 as a product of its prime factors. b Find the lowest common multiple (LCM) of 10 and 150 (1387 November 003) 1 Find the reciprocal of 3.5 Give your answer as simply as possible. 13 The number 0 can be written as m n,where m and n are prime numbers. Find the value of m and the value of n. (1387 June 005) 3 1 a Write as a power of 5 i 5 5 ii b x y 10 and x y Work out the value of x and the value of y. (1387 June 005) 15 Work out a 1.3 b c 5 3 d 1 16 a The length of the side of a square is.8 cm. Work out the area of the square. b The area of a second square is 576 cm.work out the length of one side of the square. 17 a Use your calculator to work out (6. 3.9) 1.5 Write down all the figures on your calculator display. b Put brackets in the expression so that the statement is true (Mock 003) 18 Work out the value of each of the following. Give each answer correct to one decimal place. a b c Work out the value of Write down all the figures on your calculator display. (1387 June 005) 15

16 CHAPTER 1 Number x x 0 y t 6t x 6., t.6, y is a positive number. a Work out the value of y.write down all the figures on your calculator display. b Round off your answer to an appropriate degree of accuracy. 1 p is a prime number not equal to 7 a Write down the highest common factor (HCF) of 9p and 7p x and y are different prime numbers. b i Write down the highest common factor (HCF) of the two expressions x y xy ii Write down the lowest common multiple (LCM) of the two expressions x y xy (1388 January 005) 16