Chapter 5Chapter 5

Working With NumberWorking With Number

Learning ObjectivesLearning Objectives► Recognise even/odd numbersRecognise even/odd numbers► Find multiples of a numberFind multiples of a number► Find factors of a numberFind factors of a number► Recognise prime numbersRecognise prime numbers► Write numbers in index formWrite numbers in index form► Find prime factors of a Find prime factors of a

numbernumber► Write a number as the Write a number as the

product of its prime factorsproduct of its prime factors► Find least common multiple Find least common multiple

of numbersof numbers► Find highest common factor Find highest common factor

of numbersof numbers► Find square numbersFind square numbers

► Find cube numbersFind cube numbers► Find reciprocalsFind reciprocals► Find powers of numbersFind powers of numbers► Find square roots of Find square roots of

numbersnumbers► Multiply numbers with Multiply numbers with

powerspowers► Divide numbers with powersDivide numbers with powers► Recognise surdsRecognise surds► Do calculations with surdsDo calculations with surds

Even/Odd NumbersEven/Odd Numbers

► Even numbersEven numbers End in 0, 2, 4, 6, 8End in 0, 2, 4, 6, 8

► Odd numbersOdd numbers End in 1, 3, 5, 7, 9End in 1, 3, 5, 7, 9

Example

Are the following even or odd?

2, 32, 65, 64, 345, 1234, 6543, 546, 65, 109, 100

Multiples & FactorsMultiples & Factors

► MultiplesMultiples These are the answers These are the answers

to the multiplication to the multiplication tablestables

ExamplesExamples

1)1) Write down the 1Write down the 1stst 5 5 multiples of 12multiples of 12

2)2) What is the 7What is the 7thth multiple multiple of 6?of 6?

3)3) The 5The 5thth multiple of a multiple of a number is 40. What is number is 40. What is the number?the number?

► FactorsFactors These are numbers that These are numbers that

divide in exactlydivide in exactly Best to do it in pairsBest to do it in pairs

ExamplesExamples

1)1) What are the factors of What are the factors of 12?12?

2)2) What are the factors of What are the factors of 40?40?

Prime NumbersPrime Numbers

►Prime numbers have 2 factors – itself Prime numbers have 2 factors – itself & 1& 1

►2, 3, 5, 7, 11, 13, 172, 3, 5, 7, 11, 13, 17

ExampleExample

Which of the following are prime Which of the following are prime numbers?numbers?

1, 5, 3, 7, 6, 19, 21, 25, 29, 100, 911, 5, 3, 7, 6, 19, 21, 25, 29, 100, 91

Powers/ IndicesPowers/ Indices

► Short-hand way of writing Short-hand way of writing multiplicationmultiplication

ExamplesExamples

1)1) 22×2×2×2×2×2×2×2×2×2×2×2×2

2)2) 0.10.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1×0.1

3)3) 22×2×2×3×3×3×3×3×4×4×2×2×3×3×3×3×3×4×4

Prime FactorsPrime Factors

► Put a list of prime numbers at the top of Put a list of prime numbers at the top of the pagethe page

► Keep dividing by the smallest prime Keep dividing by the smallest prime number that goes in evenlynumber that goes in evenly

EXAMPLESEXAMPLES

Find the prime factors of:Find the prime factors of:

(a)(a) 1212

(b)(b) 4040

Product of Prime FactorsProduct of Prime Factors

► Find the prime factors and write as a Find the prime factors and write as a multiplication (use powers if needed)multiplication (use powers if needed)

EXAMPLESEXAMPLESWrite the following as a product of prime Write the following as a product of prime

factors:factors:(a) 20(a) 20(b) 50(b) 50(c) 150(c) 150(d) 70(d) 70

Least Common MultiplesLeast Common Multiples

► Write as products of prime factorsWrite as products of prime factors► Include all numbers to the highest Include all numbers to the highest

powerpower

EXAMPLESEXAMPLES

Find the LCM ofFind the LCM of

(a)(a) 12 and 812 and 8

(b)(b) 45, 90 and 10545, 90 and 105

Homework for FridayHomework for Friday

1.1. Find the LCM of 12 and 66Find the LCM of 12 and 662.2. Find the LCM of 15, 39 & 45Find the LCM of 15, 39 & 453.3. Find all the factors of 18Find all the factors of 184.4. Is 15 a prime number?Is 15 a prime number?5.5. Find the prime factors of 36Find the prime factors of 366.6. Find the LCM of 8 & 12Find the LCM of 8 & 127.7. Find the 1Find the 1stst 5 multiples of 5 5 multiples of 58.8. Find all the factors of 24Find all the factors of 249.9. Find the prime factors of 24Find the prime factors of 2410.10. Find the LCM of 48, 60 & 100Find the LCM of 48, 60 & 100

Highest Common FactorHighest Common Factor

► Find the prime factors of each numberFind the prime factors of each number► Include only the numbers common to bothInclude only the numbers common to both

EXAMPLESEXAMPLES

Find the HCF ofFind the HCF of

(a)(a) 18 and 4518 and 45

(b)(b) 40 and 3640 and 36

(c)(c) 45, 90 and 10545, 90 and 105

Square NumbersSquare Numbers

► A square number is when you multiply a A square number is when you multiply a number by itselfnumber by itself

EXAMPLESEXAMPLES2222==332 2 ==442 2 ==552 2 ==662 2 ==772 2 ==

Cube NumbersCube Numbers

► A cube number is when you multiply a A cube number is when you multiply a number by itself and by itselfnumber by itself and by itself

EXAMPLESEXAMPLES2233==333 3 ==443 3 ==553 3 ==663 3 ==773 3 ==

Any PowersAny Powers

► Use the power button on your calculatorUse the power button on your calculator

xxyy or y or yxx or ^ or ^

EXAMPLESEXAMPLES2266==334 4 ==440.5 0.5 ==552.5 2.5 ==66-1 -1 ==77-1 -1 ==

Square RootsSquare Roots

► These are the opposite of square numbersThese are the opposite of square numbers► They can be written as They can be written as √ of power of ½√ of power of ½

EXAMPLESEXAMPLES

√√16 = 16 = 25250.5 0.5 ==33√125 = √125 = 33√64 = √64 =

ReciprocalsReciprocals

► In an ordinary number reciprocal In an ordinary number reciprocal means 1/means 1/

►Eg rec of 4 = , rec of 10 = , rec of 3 = Eg rec of 4 = , rec of 10 = , rec of 3 =

► In a fraction reciprocal means up-side-In a fraction reciprocal means up-side-downdown

►Eg rec of Eg rec of 22//33 = , rec of ¾ = , rec of = , rec of ¾ = , rec of 44//5 5 = =

Multiplying with IndicesMultiplying with Indices

►Consider 2Consider 233×2×244

Shortcut:Shortcut:►When we multiply numbers that are When we multiply numbers that are

the same we add the indicesthe same we add the indices

Dividing with IndicesDividing with Indices

►Consider 2Consider 277÷2÷244

Shortcut:Shortcut:►When we divide numbers that are the When we divide numbers that are the

same we subtract the indicessame we subtract the indicesaamm÷a÷ann = a = am-nm-n

ExamplesExamples

1. 21. 299×2×266

2. 2. 2299÷2÷266

3. 3. 2233×2×266×3×322×3×355

4. 4. 6666÷6÷6

5. 5. 4499×4×466÷4÷422

6. 106. 10-4-4×10×10-3-3

7. 107. 10-4-4÷10÷10-3-3

SurdsSurds

►Surds do not have exact Surds do not have exact √√

ExampleExample

Which of the following are surds?Which of the following are surds?

√√44 √25√25 √2√2 √100√100

√√1010 √36√36 √5√5 √9√9

Adding/Subtracting SurdsAdding/Subtracting Surds

► Make sure the surds are the same Make sure the surds are the same before you add/subtract thembefore you add/subtract them

ExamplesExamples

1.1. √√2 + 3√22 + 3√2

2.2. 4√3 - √34√3 - √3

3.3. 5√2 - 3√25√2 - 3√2

Simplifying SurdsSimplifying Surds

► Find the factors of the numbers and see if Find the factors of the numbers and see if you can break down some of the surdsyou can break down some of the surds

ExamplesExamples1. 1. √12√122. √752. √753. √903. √904. √84. √85. √405. √40

Dividing SurdsDividing Surds

► Split into 2 parts and do each surd Split into 2 parts and do each surd separately and then cancel at the endseparately and then cancel at the end

ExamplesExamples

1.1. √√((99//44))

2.2. √√((1010//44))

3.3. √√((2727//99))

4.4. √√((1515//1212))

Multiplying SurdsMultiplying Surds

ExamplesExamples

1. 1. √5√5××√5√5

2. √122. √12××√3√3

3. √73. √7××√14√14

4. 3√24. 3√2××√2√2

5. √25. √2×3×3√2√2