sme1 wks3-8

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Week 3

Term 1

2012

Theoretical Components

1. Read through Chapter 9 (9A & 9B). Go through

all the terminologies and examples. Make a list

of all new terms you come across. Chapter 9 is

under the Learning Brief Folder in cLc.2. Watch this only after you have studied examples from

Chapter 9:

http://www.youtube.com/watch?v=Wnc3_AekOno

3. Basic Concepts of Propositional Logic:

http://www.youtube.com/watch?v=qV4htTfow-E

4. Modus Tollens:

http://www.youtube.com/watch?v=fLlkSDb0UFk&feature

=channel

5. Modus Ponens:

http://www.youtube.com/watch?v=vtXksnrMtog

Tollens & Ponnen:

http://www.youtube.com/watch?v=s5sbEcGrdS4

6. Read through this site on what is a Logic Puzzle, how to

solve such puzzles and attempt to solve at least 2 puzzles.

Keep a record of the puzzles you have solved in your

portfolio.

http://www.logic-puzzles.org/

You may find more puzzles here:

http://www.puzzlersparadise.com/page1034.html

Practical Components1. Complete Exercises 9A and 9B from Chapter 9.

2. Attempt as many as you can, but nothing lessthan 5 exercises from here:

http://www.math.csusb.edu/notes/quizzes/tablequiz

/tablepractice.html

3. Do all the exercises from here:

http://dsearls.org/courses/M120Concepts/ClassNot

es/Logic/130B_exercises.htm

A:

Of Messi, Figo and Cantona one is honest (always tells the

truth), one is a liar (always lies), and one is ordinary (sometimes

tells the truth and sometimes lies). Deduce who is what from

the statements they make as shown below:

Messi: I am a liar.

Figo: I am ordinary.

Cantona: I am honest.

B: Investigate DeMorgans Law of negating AND and OR.

Use De Morgans laws to determine whether

the two statements are equivalent:

p q, (p q).

By the end of this week, you should be able to:

Understand the terminologies associated with the study of Propositional

Logic and Truth Tables

Use notations to represent arguments of various forms (injunction,conjunction, inverse, converse, negation, contrapositive, implication, bi-

conditional)

Determine the truth value of a statement using truth tables

Use inductive and deductive reasoning to solve logic puzzles.

Goals

Learning BriefSME1: Number

Theory, Graphs and

Networks

Negation Conjunction

Injunction Implication

Inverse Converse

Contrapositive

Syllogism Ponens

Tollens Premise

Conclusion Truth

Tables
http://www.youtube.com/watch?v=Wnc3_AekOnohttp://www.youtube.com/watch?v=Wnc3_AekOnohttp://www.youtube.com/watch?v=qV4htTfow-Ehttp://www.youtube.com/watch?v=qV4htTfow-Ehttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=vtXksnrMtoghttp://www.youtube.com/watch?v=vtXksnrMtoghttp://www.youtube.com/watch?v=s5sbEcGrdS4http://www.youtube.com/watch?v=s5sbEcGrdS4http://www.logic-puzzles.org/http://www.logic-puzzles.org/http://www.puzzlersparadise.com/page1034.htmlhttp://www.puzzlersparadise.com/page1034.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://dsearls.org/courses/M120Concepts/ClassNotes/Logic/130B_exercises.htmhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.htmlhttp://www.puzzlersparadise.com/page1034.htmlhttp://www.logic-puzzles.org/http://www.youtube.com/watch?v=s5sbEcGrdS4http://www.youtube.com/watch?v=vtXksnrMtoghttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=fLlkSDb0UFk&feature=channelhttp://www.youtube.com/watch?v=qV4htTfow-Ehttp://www.youtube.com/watch?v=Wnc3_AekOno

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ADDITIONAL

READING:http://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htm

QuizTo be completed by midnight, 26

thFeb. Quiz is available on cLc.

ForumNext Week.
http://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htmhttp://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htmhttp://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htmhttp://regentsprep.org/Regents/math/geometry/GP1/indexGP1.htm

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Week 4

Term 1

2012


1. By now, it is assumed that you have completed all

requirements of Learning Briefs for Weeks 2 & 3. If you

havent, you got to see Sheikh asap.

2. Read through Chapter 9C. Read through notes

and examples on techniques of proofs. Make

your notes. Chapter 9 is under the Learning

Brief Folder in cLc.3. Examples of Direct Method of Proof:

http://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htm

4. Examples of Indirect Proof (or Proof by Contradiction):

http://www.personal.kent.edu/~rmuhamma/Philosophy/

Logic/ProofTheory/proof_by_contradictionExamples.htm

5. Proof by Mathematical Induction:

http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=4

1373&ResourceID=155339

6. Watch these examples on methods of proof and make

your notes:


&ResourceID=155334


&ResourceID=155335

Practical Components1. Complete Exercises 9C from Chapter 9.

2. Exercise on Proof by Mathematical Induction:http://clc.act.edu.au/GroupDownloadFile.asp?Gr

oupId=41373&ResourceID=155340

Investigate how you can prove the truth of the statement: There

are exactly four prime numbers between 1 and 10, using

Exhaustion.

Find another example that can be proved by using this method.


Understand the terminologies associated with the study of PropositionalLogic and Truth Tables

Use notations to represent arguments of various forms (injunction,conjunction, inverse, converse, negation, contrapositive, implication, bi-

conditional)

Determine the truth value of a statement using truth tables

Use truth tables to determine the validity of an argument

Use various methods of proof to establish validity of arguments

Goals


Theory, Graphs and

Networks

Implications

Converse

ContrapositiveInverse

Tautology

ContradictionCounter Examples

Mathematical Induction
http://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155340http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155335http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=39726&ResourceID=155334http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://clc.act.edu.au/GroupDownloadFile.asp?GroupId=41373&ResourceID=155339http://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/proof_by_contradictionExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htmhttp://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/ProofTheory/direct_proofExamples.htm

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Additional Reading:

http://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdf

QuizNext Week

ForumDiscuss which method of proof do you like, and why.
http://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdf

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Week 5

Term 1

2012


1. Go through the notes and examples on Number

Theory (pdf version available on cLc under

Learning Brief folder.

2. Review proof by mathematical induction

3. Sharpen your skills on Logic Puzzles you find on

this link:

http://www.printable-puzzles.com/printable-

logic-puzzles.php

Additional Reading:

http://school.maths.uwa.edu.au/~gregg/Academy/200

5/inductionprobswsoln.pdf

8th

March (LINE2): I will go through Power Point Presentation

on Number Theory.so dont miss it!

There is no forum or quiz this week, so your attendance to

this class is compulsory.

If anyone has issues on Proof by Induction can ask me asmany question during this time. But, you can also come and

meet me in the Maths Staff room anytime during the week.

Practical ComponentsAttempt the questions from Exercises 4.1 to 4.5.

Practice on Logic Puzzles.Practice on Proof by Mathematical Induction.

IFYOUHAVEANYPROBLEMS(ABOUTA

NYCONCEPTSWEHAVESTUDIEDSOFA

R),YOUNEEDTOSEESHEIKHASAP.

NO INVESTIGATION FOR THIS WEEK.

PREPARE YOURSELF FOR THE IN-CLASS ON LOGIC

PUZZLES AND PROOF BY INDUCTION.

Quiz Next Week


Use integers and their properties to understand basic principles ondivisibility, greatest common divisors, least common multiples and

modular arithmetic

Review logic puzzles and proof by mathematical induction for theassessment.

NOTE: We are meeting inRoom 23 on Thursday 8

th

March (LINE 2) to discuss

the assessment task for

next week. Your

attendance to this meeting

is compulsory!

Goals


Theory, Graphs and

Networks

For

um

Next Week.

Prime NumbersDivisibility

Euclidean Algorithm

Useful LINK:

http://www.onlinemathlearnin

g.com/mathematical-induction-examples.html


g.com/divisibility-rules-

explained.html
http://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.php

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Week 6

Term 1

2012





2. Divisibility RULES:

http://www.youtube.com/watch?v=i16N01IdIh

k&feature=topics

http://www.youtube.com/watch?v=y1rVfa1nhj

w

http://www.youtube.com/watch?v=P5oHmgB4

Nfs&feature=channel

3. Review Truth Tables & proof by mathematical

induction

4. Sharpen your skills on Logic Puzzles you find on

this link:

http://www.printable-puzzles.com/printable-

logic-puzzles.php

Additional Reading:

INDUCTION:

http://school.maths.uwa.edu.au/~gregg/Academy/200

5/inductionprobswsoln.pdf

INTEGER DIVISIBILITY:

http://www.cs.cmu.edu/~adamchik/21-

127/lectures/divisibility_1_print.pdf

Practical ComponentsAttempt the questions from Exercises 4.1 to 4.5.

Practice on Logic Puzzles.Practice on Proof by Mathematical Induction.

IFYOUHAVEANYPROBLEMS(ABOUTA

NYCONCEPTSWEHAVESTUDIEDSOFA

R),YOUNEEDTOSEESHEIKHASAP.

Construct modulo 6 addition and multiplication tables.

Quiz No Quiz this week, so prepare for yourassessment.


Use integers and their properties to understand basic principles ondivisibility, greatest common divisors, least common multiples and

modular arithmetic

Review logic puzzles and proof by mathematical induction for theassessment.

NOTE: In-Class Assessment

is on Thursday, 15th

March.

This will be during LINE 2 @

8.45 a.m. in Room 23.

You can bring handwritten

notes. Be prompt.

Goals


Theory, Graphs and

Networks

For

um

No Forum this week, so

prepare for your assessment.

Useful LINK:


g.com/mathematical-induction-examples.html


g.com/divisibility-rules-

explained.html
http://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/divisibility-rules-explained.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.onlinemathlearning.com/mathematical-induction-examples.htmlhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://www.cs.cmu.edu/~adamchik/21-127/lectures/divisibility_1_print.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://school.maths.uwa.edu.au/~gregg/Academy/2005/inductionprobswsoln.pdfhttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.printable-puzzles.com/printable-logic-puzzles.phphttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topics

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Week 7

Term 1

2012





2. Divisibility RULES:

http://www.youtube.com/watch?v=i16N01IdIh

k&feature=topics

http://www.youtube.com/watch?v=y1rVfa1nhj

w

http://www.youtube.com/watch?v=P5oHmgB4

Nfs&feature=channel

3. Euclidean Algorithm for gcd:

http://www.youtube.com/watch?v=fwuj4yzoX1o

4. Modular Arithmetic Notes & Examples & Exercises:

http://www.dsert.kar.nic.in/textbooksonline/Text%20bo

ok/English/class%20x/maths/English-Class%20X-Maths-

Chapter06.pdf

Practical Components

Attempt the questions from Exercises in the pdffile available under 4.

Read the notes below, and answer the questions

that follow.

Quiz No Quiz this week.


Use integers and their properties to understand basic principles on

divisibility, greatest common divisors (gcd), least common multiples (lcm)

and modular arithmetic

Use Euclidean Algorithm to find the gcd of a pair of numbers

Recognise congruent elements Find the residues in a modular system

Construct Cayleys table and appreciate the importance the modular

system

USE CLASSPAD TO COMPUTE GCD, AND PERFORM MODULAR

ARITHMETIC

This is a busy week,

manage your timewisely, and dont

wait but see me if

you are stuck on any

concept.

Goals


Theory, Graphs and

Networks
http://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=fwuj4yzoX1ohttp://www.youtube.com/watch?v=fwuj4yzoX1ohttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.dsert.kar.nic.in/textbooksonline/Text%20book/English/class%20x/maths/English-Class%20X-Maths-Chapter06.pdfhttp://www.youtube.com/watch?v=fwuj4yzoX1ohttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=P5oHmgB4Nfs&feature=channelhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=y1rVfa1nhjwhttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topicshttp://www.youtube.com/watch?v=i16N01IdIhk&feature=topics

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The Sieve of EratosthenesHow many prime numbers are there from 1 to 100? The following procedure is a relatively simple way of

identifying the primes to 100.

1. Put a single slash (/) through the 1 block with a blue crayon or colored pencil. One is special:It is the unit.

2. Circle 2 in orange. Then cross out with red all other numbers in the chart divisible by 2.

(Another way of saying this is: Cross out all numbers in the chart that are multiples of 2. It may help some

children to have them count by twos.)

3. Circle in orange the next prime number: 3. With red, cross out any other multiple of 3 that has not

already been crossed out. (It may help some children to count by threes and cross out any of these

numbers not already crossed out.)

4. Circle in orange the next prime number: 5.

With red, cross out any other multiple of 5 that has not already been crossed out. It may help some

children to encourage them to count by fives and cross out any numbers not already crossed out.)

5. Continue in this manner until all numbers are circled in orange (the primes) or crossed out in red (the

composites). Note for what prime number you did not have to cross out any multiples to 100. Why was itunnecessary to cross out any numbers for this prime? Will it be necessary to check for multiples of the

remaining primes or can you simply circle in orange all remaining numbers at this point?

Forum On cLc.The Math Book from Hell had the following rather unrealistic fraction question:

What is the sum of 1/54 and 1/72 of a kilometre? Rodney sensed that 54 x 72

(3888) was not the lowest common denominator and that using it would mean a

lot of extra work.(a) How could the lowest common denominator for 1/54 and 1/72 be found?

(b)What is the LCM of these fractions?

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Modular Arithmetic and Congruence

CONGRUENCE

We say that two integers aand b are congruent modulo n if and only if

n|(a-b), where n is a positive integer.

This is written as: ()

Example 1

Are 25 and 17 congruent modulo 4?

25 17 = 8. 4|8, 25 17( 4)

Properties:

, ,, ,, , +:

1. () =

2. () (

0).

3. (). This is known as the Reflexive Property

4. () () . This is known as the

Symmetric Property

5. () () (). This is

known as the Transitive Property.

6. () + + () ()

7. () () +

+ () ()

8. () ()

Example 2

Prove congruence Property 5. () () ()

() (), |( )|( )From the division identity, we have:( ) = 1( ) = 2 1,2 , = 0

( ) + ( ) = 1 + 2 = (1 + 2) = 1 + 2

|( ), ()Q.E.D.

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The basic congruency properties can be regarded as rules for modular

arithmetic, which operates with equality (=) replaced by congruence ().Properties 6 & 7 show that multiplication preserve congruency, but the

operation is not reversible. This is obvious because division does not

always give an integral answer. These properties allow some problems tobe solved quite easily.

Example 3

Find the remainder when 330

is divided by 7.

33 = 27 1( 7)

2710 (1)10( 7) 8

330 1( 7)

From Property 2, the remainder when 3

30

is divided by 7 is 1.

Exercises:

1. Prove Property 6

2. Prove Property 7.

3. Find the remainder when:

i) 230 is divided by 7

ii) 516 is divided by 24

iii) 9120 is divided by 40

iv) 220 is divided by 41

v) 2316 is divided by 7

4. Use the remainder when 330 is divided by 7 to show that 330-1 is

divisible by 7.

5. Show that 7|(36 1) +

6. Show that 24|(52 1) +

7. Show that 41|(220 1) +

8. Show that if () and m|n, then ()

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Week 8

Term 1

2012


You will find the reading/examples/exercises for this

week below.

Practical ComponentsRead the notes and examples and attempt a few

questions.

Quiz On cLc.


Appreciate the use of Diophantine Equations to find integer solutions to

linear equations

Determine which of the linear equations are solvable with integer solution

Solve a Diophantine Equation using Euclidean Algorithm to find particular

solution. Solve any equation of the form + = , given | =

gcd(, ) to find all solutions (if they exist).

By now you should

now these:o gcd, lcm using prime factors

o Cayleys Table

o Euclidean Algorithm

o Congruence

Use this as checklist!

Got a question = see Sheikh

Goals


Theory, Graphs and

Networks

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Investigate the use of Euclidean Algorithm to find a particular solution to a Diophantine Equation.

This video may be helpful:

http://www.youtube.com/watch?v=FjliV5u2IVw

Remember, this video shows how to use Euclidean Algorithm to find a particular solution.

Do a detailed write-up of this technique, with a different example (there is no limit, so you can write as

many examples).

Forum Next week.
http://www.youtube.com/watch?v=FjliV5u2IVwhttp://www.youtube.com/watch?v=FjliV5u2IVwhttp://www.youtube.com/watch?v=FjliV5u2IVw

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