Chap01 - 001-016

8/3/2019 Chap01 - 001-016

1/16

1

1.1 Properties of whole numbersA factor of a number,x, is a number which divides intox an exact number of times.So 3 is a factor of 12 because 3 4 123 and 4 are called a factor pair of 123 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 byy.The first three multiples of 5 are 5, 10 and 1521 is not a multiple of 5 because 21 leaves a remainder when divided by 5

Find all the factors of 20 Write them in factor pairs.

Solution 11, 202, 104, 5

A number which has exactlytwo factors is called a prime number.So, 2, 3, 5 and 23 are prime numbers but 1, 4 and 15 are not.1 is not a prime number because it only has one factor, 1

2 is the only even prime number. Explain why.

Solution 22 is a prime number because it has exactly two factors, 1 and 2

All other even numbers have at least 3 factors, 1, 2 and the number itself.

A common factor of two numbers,x andy, is a number which is both a factor ofx and is also afactor ofy.

So 4 is a common factor of 12 and 203 is not a common factor of 12 and 20 because 3 is not a factor of 20

Find all the common factors of 12 and 20

Solution 31, 2, 3, 4, 6 ,12

1, 2, 4, 5, 10, 20

1, 2, 4 are the common factors of 12 and 20

Example 3

Example 2

Example 1

1C H A P T E R

Number

In each case the productof the two numbers is 20

These are the factors of 12

These are the factors of 20

These three factors appear in both lists.

8/3/2019 Chap01 - 001-016

2/16

Exercise 1A

1 Which of the following numbers are factors of 18?

a 12 b 6 c 9 d 3 e 36

2 Which of the following numbers are factors of 30?

a 1 b 20 c 15 d 3 e 6

3 Find all the factors of 50 Write them in factor pairs.

4 List all of the factors of the following numbers.

a 8 b 10 c 16 d 24 e 28

f 32 g 36 h 40 i 60 j 100

5 List all the common factors of

a 6 and 8 b 6 and 9 c 6 and 10 d 8 and 12

e 12 and 15 f 10 and 20 g 15 and 20 h 18 and 24

6 a Write down the first three multiples of 4

b Write down the first three multiples of 10c 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 23

7 State whether the following statements are true or false.

a 12 is a multiple of 2 b 14 is a factor of 7

c 24 is a multiple of 3 d 72 is a multiple of 9

e 12 is both a multiple of 6 and a factor of 36 f 9 is a factor of 27

g 6 is a multiple of 1 h 4 is a multiple of 12

8 Show that 33 is a factor of 3003

9 Find the first multiple of 29 which is greater than 2000

10 Find two prime numbers between 110 and 120

11 Bertrands theorem states that Between any two numbers n and 2n, there always lies at leastone prime number, providing n is bigger than 1.Show that Bertrands theorem is true i for n 10 ii for n 34

12 Find a number which has exactly

a 4 factors b 3 factors c 7 factors d 10 factors

1.2 Multiplication and division of directed numbersA directed number is a number with a or a sign. 4 and 3 are examples of directed numbers.Often a directed number is written in brackets, for example (4) and (3).

Multiplication

(3) is the same as 3 (2) is the same as 2

so (3) (2) is the same as 3 2 6

3 2 also means 2 2 2 3 (2) means (2) (2) (2)so 3 (2) or (3) (2) (6)

2

CHAPTER 1 Number

8/3/2019 Chap01 - 001-016

3/16

Look at the patterns in the multiplications on the right.

The blue numbers are decreasing by 1

The orange numbers are increasing by 2

The pattern continues like this.

The rules are() () () positive positive positive() () () positive negative negative() () () negative positive negative() () () negative negative positive

Division

3 2 6 6 3 2 and 6 2 3

(3) (2) (6), so (6) (3) (2) and (6) (2) (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 followingif the signs are the same, the answer is positiveif the signs are different, the answer is negative.

a Work out (5) (3) b Work out (16) (2)

Solution 4a (5) (3) (15)

(15) is also written as 15

b (16) (2) (8)

(8) is also written as 8

Exercise 1B

1 Work outa (2) (4) b (3) (5) c (4) (6)

d (3) (5) e (2) (5) f (4) (5)

g (3) (8) h (1) (9) i (4) (4)

2 Work out

a (6) (3) b (8) (4) c (10) (5) d (12) (3)

e (8) (4) f (12) (12) g (14) (2) h (12) (4)

3 Find the missing directed number.

a (10) ( ) (2) b (8) ( ) (2) c (3) ( ) (12)

d (5) ( ) (20) e (5) ( ) (25) f ( ) (4) (20)g ( ) (3) (4) h ( ) (4) (5) i (16) ( ) (2)

The signs are the same so the answer ispositive and 16 2 8

The signs are different so the answer isnegative and 5 3 15

Example 4

3

1.2 Multiplication and division of directed numbers CHAPTER 1

(3) (2) (6)

(2) (2) (4)

(1) (2) (2)

0 (2) 0

(1) (2) (2)

(2) (2) (4)

(3) (2) (6)

8/3/2019 Chap01 - 001-016

4/16

4 Work out the product of

a (6) and (3) b (5) and ( 4) c (3) and (5) d (6) and (6)

e (3) and (4) f (4) and (9) g (5) and (4) h (3) and (2)

1.3 Squares and cubes

The square of a number,x, is the number which is the productx x.The square of the numberx is writtenx2.So the square of 10 is written as 102 10 10The square of 10 is 100

The cube of a number,y, is the number which is the producty y y.The cube of the numbery is writteny3.So the cube of 4 is written 43 4 4 4The cube of 4 is 64

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 4 since 42

16Since (4)2 16, the negative square root of 16 is 4It 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 31000, is 10 since 103 1000

It is possible to find the cube root of a negative number.

Work out 23 3

27

Solution 523 8

3

27 3

23 3

27 5

Exercise 1C

1 Work out

a 32

b 52

c 112

d 132

e 152

f 1002

2 Work out

a (2)2 b (4)2 c (10)2 d (12)2

3 Work out

a 33 b 13 c 53 d (10)3 e (4)3

4 Work out

a 4 32 b 52 100 c 23 32 d 3

8 42

e 31000 100 f 43 23 g (1)3 23 (3)3

h 42 (3)3 i 3125 81 j 6

2

2

2

Example 5

4

CHAPTER 1 Number

2 2 2 8

because (3) (3) (3) 27

8 (3) 5

8/3/2019 Chap01 - 001-016

5/16

5 Here is a number pattern

12 13

(1 2)2 13 23

(1 2 3)2 13 23 33

Show that the next line of the number pattern is also true.

1.4 Index lawsAs well as squares and cubes it is possible to represent a number multiplied by itself any number oftimes. For example,

24 (2 raised to the power 4) means 2 2 2 2

36 (3 raised to the power 6) means 3 3 3 3 3 3

Another name for power is index.

Work out a 34 b 26

Solution 6

a 34 3 3 3 3 81

b 26 2 2 2 2 2 2 64

To work out one number raised to a power multiplied by the same number raised to a second poweryou add the powers.

For example 23 24 27 because 23 2 2 2 and 24 2 2 2 2and (2 2 2) (2 2 2 2) 234 27

To divide one number raised to a power by the same number raised to a second power you subtractthe powers.

For example, 36 34 cancelling all the 3s

on the bottom with four of the 3s on the top.

So 36 34 364 32

a Work out 24 25. Give your answer as a power of 2

b Work out 58 55. Give your answer as a power of 5

c Work out (32)4. Give your answer as a power of 3

d Work out 4 47. Give your answer as a power of 4

Solution 7

a 24 25 245 29 b 58 55 585 53

c (32)4 32 32 32 32 32222 38 d 4 47 417 48

Example 7

3 3 3 3 3 3

3 3 3 3

3 3 3 3 3 3

3 3 3 3

Example 6

5

1.4 Index laws CHAPTER 1

8/3/2019 Chap01 - 001-016

6/16

Work out74

7

8

76

Solution 8

74

7

8 76

77

1

8

0

72 49

Exercise 1D

1 Write as a power of 2

a 24 25 b 23 24 c 22 26 d 24 23 e 24 26


a 34 32 b 35 32 c 34 3 d 36 32 e 310 34

3 Write as a power of a single number

a 44 42 b 57 52 c 34 32 d 64 63 e 104 102

4 Find the value of n

a 3n 32 33 b 85 8n 82 c 25 2n 210 d 3n 35 39 e 26 23 2n

5 Work out

a 34 32 b 45 43 c 25 22 d 104 102 e 65 65


a 33

3

4

35 b (33)2 c

3

3437 d

343

9

33 e

3

3

2

2

3

3

1

5

0

7 Write as a power of a single number

a 23

2

5

24 b

34

3

4

33 c

53

5

6

55 d

108

1

07103 e

45

4

2

4

8 Work out

a 52

5

5

52 b

323

4

32 c

424

7

43 d

2

2

3

4

2

2

4

2 e

3

34

3

3

7

2

9 Work out the value of n in the following.

a 40 5 2n b 32 2n c 50 5n 2 d 48 3 2n e 54 2 3n

1.5 Order of operationsSome expressions include powers and other operations.BIDMAS gives the order in which operations should be carried out.

Remember that BIDMAS stands for

BracketsIndicesDivisionMultiplicationAdditionSubtraction

Example 8

6

CHAPTER 1 Number

Work out means evaluate the expression ratherthan leaving the answer as a power of 7

8/3/2019 Chap01 - 001-016

7/16

8/3/2019 Chap01 - 001-016

8/16

8/3/2019 Chap01 - 001-016

9/16

8/3/2019 Chap01 - 001-016

10/16

10 Work out, giving your answers correct to one decimal place

a6.32

7

.5

3.32 b

2.5

83

0

.6

10 c

3.53

2

.6

8.5 d

8.7

6.5

30

e f 5.

2

5

.

2

22

1.

5

52 g

4

3

.

.

5

4

3

2

1

1

8

0 h 14.6

3

2

.

.

9

6

3

i j

11 Find the reciprocal of each of the following numbers.

a 4 b 8 c 40 d 0.625 e 3.2

1.7 Prime factors, HCF and LCMA prime factor of the number n, is a prime number which is a factor of n.

The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

The prime numbers in this list are 2, 3 and 5So the prime factors of 30 are 2, 3 and 5

Prime numbers can be thought of as the basis of all whole numbers because all whole numbers areeither prime or can be written as a product of prime numbers.

For example, 15 is not prime, but can be written as the product 3 5

12 is not prime, but can be written as the product 2 2 3

For small numbers it is easy to see what prime numbers to use.For larger numbers use the following method.

Write 72 as

a the product of its prime factors b the product of powers of its prime factors.

Solution 13

a The prime factors of 72 are 2 and 3

2 72

2 36

2 18

3 9

3 3

1

72 2 2 2 3 3

b 72 23 32

The highest common factor (HCF) of two numbers is the largest number which is a factor of bothof the numbers.

For example, the highest common factor (HCF) of 8 and 12 is 4 because it is the largest number thatis a factor of both 8 and 12

For larger numbers, it is useful to list the factors of each number and then pick out the largestnumber that appears in all the lists.

Example 13

3.63 4 203.63 4 20

6.42 20

20 30

17.4 2.42

4.5

0

CHAPTER 1 Number

Divide 72 by 2

Divide 36 by 2

Divide 18 by 2

Divide 9 by 3

Divide 3 by 3

8/3/2019 Chap01 - 001-016

11/16

Find the highest common factor (HCF) of 24 and 36

Solution 14The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36

The numbers which appear in both lists, that is the common factors, are 1, 2, 3, 4, 6 and 12So 12 is the highest common factor of 24 and 36

The lowest common multiple (LCM) of two numbers is the smallest number which is a multiple ofboth numbers.

For example, the lowest common multiple of 8 and 12 is 24 because it is the smallest number whichis a multiple of both 8 and 12

For larger numbers, it is useful to list the multiples of each number and then pick out the smallestnumber that appears in both lists.

Find the lowest common multiple (LCM) of 15 and 20

Solution 15The first few multiples of 15 are 15, 30, 45, 60, 75 ,The first few multiples of 20 are 20, 40, 60, 80, 100,

The smallest number which appears in both lists is 60So the lowest common multiple of 15 and 20 is 60

The HCF can be worked out for large numbers if each of the numbers is written as a product of itsprime factors.

For example, for the numbers 120 and 144 the products are

So 2 2 2 3 24 is the highest common factor of 120 and 144

In terms of products of powers of their prime factors,

Their highest common factor 24 ( 23 3) is the product of the lowest power of each of theircommon prime factors.

Find the highest common factor (HCF) of 750 and 225

Solution 16750 2 3 53

225 32 52

HCF 3 52 75

Example 16

120 23 3 5144 24 32

120 2 2 2 3 5144 2 2 2 2 3 3

Example 15

Example 14

11

1.7 Prime factors, HCF and LCM CHAPTER 1

Write 750 and 225 as the product of powers of their prime factors.

The common prime factors are 3 and 5The lowest power of 3 is 1 (as 3 31) and the lowest power of 5 is 2

8/3/2019 Chap01 - 001-016

12/16

To find the LCM of 36 and 120, list the multiples of 36 and 120 until the same multiple appears inboth lists.

The multiples of 36 are 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, The multiples of 120 are 120, 240, 360, The LCM of 36 and 120 is 360As a product of its prime factors, 360 2 2 2 3 3 5

As a product of their prime factors the numbers 36 and 120 are

The LCM of 36 and 120 (360) is the product of the common prime factors andall other primefactors, that is 2 2 3 3 2 5

In terms of products of powers of their prime factors,

Their lowest common multiple, 360 ( 23 32 5), is the product of the highest power of all their

prime factors.

Find the lowest common multiple (LCM) of 750 and 225

Solution 17750 2 3 53

225 32 52

LCM 2 32 53 2250

Exercise 1G

1 Find the two prime numbers that are between 30 and 40

2 Find two prime numbers which have a sum of 7

3 Find two prime numbers which have a product of 14

4 Find two prime numbers which are factors of 20



7 Write the following numbers as a product of two prime factors.

a 10 b 15 c 21 d 22 e 33 f 39

8 Which of the following show a number written correctly as a product of prime factors?

a 12 2 2 3 b 18 2 9 c 20 2 2 5

d 16 2 2 2 2 e 56 2 ,2 ,2 ,7 f 10 2 5 1

9 Write each of these numbers as a product of its prime factors.

a 30 b 42 c 48 d 36 e 60 f 63g 54 h 80 i 76 j 88 k 68 l 66

Example 17

36 22 32

120 23 3 5

36 2 2 3 3120 2 2 2 3 5

2

CHAPTER 1 Number

Write 750 and 225 as the product of powers of their prime factors.

The highest power of 2 is 1 (as 2 21)The highest power of 3 is 2The highest power of 5 is 3

8/3/2019 Chap01 - 001-016

13/16

13

1.7 Prime factors, HCF and LCM CHAPTER 1

10 Find the highest common factor (HCF) of the following pairs of numbers.

a 12 and 14 b 6 and 9 c 6 and 8 d 8 and 10 e 6 and 10

11 Find the highest common factor (HCF) of the following pairs of numbers.


12 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 12 e 10 and 15

13 Find the lowest common multiple (LCM) of the following pairs of numbers.


14 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 100

15 Frank has two flashing lamps.The first lamp flashes every 4 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, 360 2 2 2 3 3 5Write 720 as a product of its prime factors.

17 The number 48 can be written in the form 2n 3 Find the value of n.

18 The number 189 can be written in the form 3n p where n andp are prime numbers.Find the value of n and the value ofp.

19 The number 120 can be written in the form 2n m p where n, m andp are prime numbers.Find the value of each of n, m andp.

20 x 2 32 5,y 23 3 7

a Find the highest common factor (HCF) ofx andy.

b Find the lowest common multiple (LCM) ofx andy.

21 2 3 5 7 21 22 24

a Which of the numbers in the list are factors of 288?

b Which of the numbers in the list are factors of 550?

22 Write each of these numbers as a product of its prime factors.a 105 b 539 c 231

d 847 e 1001

23 Find the lowest common multiple of these pairs of numbers.

a 24 and 30 b 27 and 36 c 28 and 35

d 36 and 42 e 54 and 72

24 Find all the integer values of n less than or equal to 10 for which 2n 1 is a prime number.

25 a Any square number which is even is always a multiple of 4. Explain why.

b Investigate what the corresponding answer is for square numbers which are odd.c Explain why the number 1050 3 cannot be a square number.

8/3/2019 Chap01 - 001-016

14/16

Chapter summary

Chapter 1 review questions1 Work out

a 402 b 70002 c 100d 4900

2 Work out

a 23 b 103 c 53 d 13 e 303

3 Work out

a 42 25 b 23 23 c 33 23 d 53 103

4 Work out

a 3 25 b 4 36 c 10 9 d 8 49

5 Work outa 42 8 b 43 82 c 103 102 d 33 3

4

CHAPTER 1 Number

You should now know that:

the power or index of a number shows how many of the number are multiplied together, forexample, in 25 the 5 is the power or index and five 2s are multiplied together, that is

2 2 2 2 2the 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 givesthe 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 factorof both of the numbers

the lowest common multiple (LCM) of two numbers is the smallest number which is amultiple 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 multipliedtogether

use the rule for subtracting powers when one number raised to a power is divided by the samenumber 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.

#

#

#

#

#

#

#

#

#

#

#

#

#

#

8/3/2019 Chap01 - 001-016

15/16

6 Work out

a 52 b 102 c (9)2 d (5 1)2 e (2 3)2

7 Write each of the following as a single power of 2

a 23 24 b 26 24 c (23)2 d 2 28 e 26 2

8 Write each of the following as a single power of 3

a 3

3234 b

32

3

4

33 c

3

3

5

2

3

3

4

3

9 Each of the following represents a number written as a product of powers of its prime factors.Find the number.

a 23 32 b 2 33 5 c 23 3 7

10 a Express 108 as a product of powers of its prime factors.

b Find the highest common factor (HCF) of 108 and 24 (1387 June 2004)

11 a Express 120 as a product of its prime factors.

b Find the lowest common multiple (LCM) of 120 and 150 (1387 November 2003)

12 Find the reciprocal of 3.5 Give your answer as simply as possible.

13 The number 40 can be written as 2m n, where m and n are prime numbers.Find the value of m and the value of n. (1387 June 2005)

14 a Write as a power of 5i 54 52 ii 59 56

b 2x 2y 210 and 2x 2y 24Work out the value ofx and the value ofy. (1387 June 2005)

15 Work out

a 1.32 b 13.69 c 253 d 14

16 a The length of the side of a square is 4.8 cm. Work out the area of the square.

b The area of a second square is 576 cm2. Work out the length of one side of the square.

17 a Use your calculator to work out

(6.21

.253.9)2

Write down all the figures on your calculator display.

b Put brackets in the expression so that the statement is true.

14.5 2.6 4 .5 3.6 49.95 (Mock 2003)

18 Work out the value of each of the following. Give each answer correct to one decimal place.

a14.7

2

.5

21.2 b

18

2

.

.

7

5

2

3

1

.7

.8 c

20

1

0

2

.5

8.62

19 Work out the value of 3.82 75Write down all the figures on your calculator display. (1387 June 2005)

15

Chapter 1 review questions CHAPTER 1

8/3/2019 Chap01 - 001-016

16/16

20 y2

x 6.4, t 4.6,y is a positive number.

a Work out the value ofy. Write down all the figures on your calculator display.

b Round off your answer to an appropriate degree of accuracy.

21 p is a prime number not equal to 7

a Write down the highest common factor (HCF) of 49p and 7p2

x andy are different prime numbers.

b i Write down the highest common factor (HCF) of the two expressionsx2y xy2

ii Write down the lowest common multiple (LCM) of the two expressionsx2y xy2 (1388 January 2005)

x2 4x

2t2 6t

CHAPTER 1 Number

Documents

Chap01 - 001-016