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C.
TEMPLATE STRUCTURE
Modules must follow the template pattern in both structure and terminology, and
must include all the items listed below.
I.
INTRODUCTION
1. Title of Module
Module Developer Writing Tip. The title of the module is provided by AVU.
Mathematics 2, Number Theory, Paul Cheqe, Amoud University
*One relevant image must be inserted here.
2. Prerequisite Courses or Knowledge
Module Developer Writing Tip., Module Developers should specify the prerequisite courses or knowledge required in order
for learners (preservice teachers) to start the module.
Basic Mathematics
3. Time
Module Developer Writing Tip. In this section, Module Developers should approximate how much time (total in hours) is
required to complete the module.
120 hours
4. Material
Module Developer Writing Tip. In this section, Module Developers should specify the materials required to complete the
module.
Study book online or on CD; ICT activity files online or on CD, References online,
Pre-assessment materials,Necessary freely distributed software
*One relevant image must be inserted here.
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5. Module Rationale
Module Developer Writing Tip. In this section, Module Developers should specify the main rationale for the module. A good
rationale will clearly present why learners (preservice teachers) will better teach the subject matter having completed the
module (maximum length : 50-75 words).
Number theory is an essential module that assists teachers in understanding and
interpreting the properties of numbers. It forms a background to the numerous proofs
and solutions of various mathematical equations. It forms the backbone of theory
crucial for the teaching of secondary school mathematics and is an important
building block in the study of higher-level mathematics.
*One relevant image must be inserted here.
II.
CONTENT
6. Overview
Module Developer Writing Tip. In this section, Module Developers should first write an OVERVIEW of the module. An
overview briefly presents the content of a module in paragraph form (maximum length : 100-150 words). The Module
Developers will also be briefly interviewed (videotaped) to present the overview of the module. This video file (Quicktime, .mov)
will be included with this section of the module.
Second, the overview must also be accompanied by a clear OUTLINE of the content. Contrary to an overview, an outline is not
a continuous text, it presents the content of the module in point form, and includes the approximate time required to complete
each unit, element or theme (the total time must be equal to the time indicated in section 3 above.. An outline could take the
form of a table of contents for the module.
Third, Module Developers should draw a GRAPHIC ORGANIZER - a graphical way of organizing information so it can be
better understood and retained. They are powerful tools in open and distance education that can be used to enhance learning.
For more information on graphic organizers, see Annex 1 or consult eduplace (http://www.eduplace.com/graphicorganizer/) for
examples of graphic organizers (Annex 2).
*One relevant image must be inserted here.
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Overview
The number theory module consists of two units. It pre-supposes the teacher trainee
is conversant with Basic mathematics. The first unit deals with properties of integers
and linear diophantine equations. It progresses from properties of integers through
divisibility with remainder, prime numbers and their distribution, Euclid’s proof of
infinitely many primes and Euclid’s algorithm and its application in solving linear
diophantine equations. The unit is concluded with Pythagorean triplets and Fermat’s
last theorem for the vitas powers and the proof of Wiles
The second unit assumes unit one as a prerequisite for the trainees. It introduces the
field of integers( mod p), squares and quadratic residues, Alar’s criterion, Legendre
symbol, Gauss lemma and quadratic reciprocity law, Euclid’s algorithm and unique
factorisation of Gaussian integers, arithmetic of quadratic fields and application of
diophantine equations, and is concluded with Fermat’s last theorem for cubes, Pell’s
equation and units in real quadratic fields.
*One relevant image must be inserted here.
Outline
Unit 1: Properties of integers and linear Diophantine equations
Level 1. Priority B. Basic Mathematics 2 is prerequisite.
Properties of integers. Divisibility with remainder. Prime numbers and their
distribution. Euclid’s proof of infinitely many primes. Euclid’s algorithm.
Consequences, residue classes, the integers (mod n). The case of prime n. Primitive
roots and indices. Use of Euclid’s algorithm in solving linear Diophantine equations.
Pythagorean triplets and Fermat’s last theorem for the vitas powers.
Unit 2: Theory of congruences and quadratic fields
Level 2. Priority B. Number Theory 1 is prerequisite.
The field of integers (mod p). Squares and quadratic residues. Alar’s criterion.
Legendre symbol. Gauss lemma and quadratic reciprocity law. Evaluation of
quadratic character by the reciprocity law. Quadratic fields. Norm and trace. Euclid’s
algorithm and unique factorisation of Gaussian integers. Arithmetic of quadratic
fields and application of Diophantine equations. Fermat’is last theorem for cubes.
Pell’s equation and units in real quadratic fields.
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Graphic Organizer (can be drawn manually)
Diophantine
Equations
Euclid’s
Algorithm
Integers mod p,
Squares and
quadratic
residues
Alar’s criterion,
legendre
symbol, gauss
Lemma
Quadratic fields
and reciprocity
Law
Fermat’s last
theorem
Pythagorean
Triplets
INTEGERS
Divisibility, prime
numbers,
primitive roots
and indices
Norm and Trace
Factorisation of
Gaussian
Integers
Application of
Diophantine
and Pell’s
Equation
7. General Objective(s)
Module Developer Writing Tip. In this section, Module Developers should outline the general objective(s) of the module, as
specified in the curriculum. Writing clear, informative, concise and understandable objectives is extremely important in an Open
and Distance Learning Module.
The trainee is equipped with knowledge of the properties of numbers and the relationships
among numbers necessary to effectively teaching mathematics in secondary schools.
*One relevant image must be inserted here.
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8. Specific Learning Objectives (Instructional Objectives)
Module Developers Writing Tip. In this section, Module Developers should identify the specific learning objectives for each
unit, element or theme of the module. Each specific objective will be at the heart of a teaching and learning activity. Writing
clear, informative, concise and understandable objectives is extremely important in an Open and Distance Learning Module.
Clear, concise, informative and well-written objectives help learners organize their efforts. Objectives provide some basis and
guidance for the selection of instructional content and procedures. Finally, they also help the instructor (or evaluator) assess
the extent to which a learner masters an objective.,.
Many researchers present four characteristics essential to writing learning objectives:
- A good learning objective is clear and concise, leaving little room for interpretation.
- A good learning objective always states what a learner is expected to achieve.
- A good learning objective describes, when possible, the conditions under which a learner must perform a task.
- A good learning objective, when possible, clarifies how well the student must perform a task, in order for the performance to
be acceptable.
Attached, is Bloom’s taxonomy (Annex 3). It provides useful verbs that can be used to write learning objectives which relate
either to cognitive, affective, or psychomotor domains.
Unit 1:
By the end of this unit, the learners should be able to:
Have a knowledge and understanding of the basic concepts related to and
the properties of numbers
Have knowledge and understanding of the relationships and recurring
patterns among numbers.
Illustrate the properties of Integers and divisibility with remainders
Compute the greatest common divisor and least common multiple by
factorisation
Compute the greatest common divisor using Euclidean algorithm
Illustrate properties of prime numbers and their distribution
Illustrate Euclid’s proof of infinitely many primes
Evaluate integers (mod n), the case of prime n, primitive roots and indices
Use and apply Euclid’s algorithm in solving linear Diophantine equations of
the nature mx  ny  k
Illustrate Pythagorean triplets and Fermat’s last theorem
Analise pythagorean triplets and Fermat’s last theorem.
Unit 2:
By the end of this unit, the learners should be able to:
-
Illustrate the field of integers mod p, squares and quadratic residues
Outline Euler’s criterion
Use Legendre symbol, Gauss Lemma and quadratic reciprocity law
Evaluate quadratic character by the quadratic reciprocity law
Define the Norm
Apply Eucliud’s algorithm in the factorisation of Gaussian integers
Explore arithmetic of quadratic fieldsand application of Diophantine equations
Investigate Pell’s equation and units in real quadratic fields.
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III.
TEACHING AND LEARNING ACTIVITIES
9. Pre-assessment
Module Developer Writing Tip. It is generally agreed that there are three important types of assessments in Open and
Distance Learning :
- pre-assessment ;
- formative assessment ;
- summative assessment.
In this Module Developer Writing Tip, we will discuss pre-assessments.
Pre-assessments are required to highlight the learner’s mastery level of the learning objectives. A pre-assessment always has
more to do with helping students learn than with underlining their mistakes. Pre-assessments allow both the learner and the
instructor to determine what is already known by the learner in a specific domain. Pre-assessment is also critical to recognize
prior knowledge so learners can engage fully in the learning activity in order to construct new knowledge appropriate to their
level.
Here are some of the benefits of pre-assessing as identified by Kingore (2004)1:
- Pre-assessment can motivate students to be more involved in, and attentive to, instruction and learning experiences as the
pre-assessment helps more clearly identify what they know and what they need to know.
- Pre-assessment appropriately raises the learners’ level of concern by signaling what they need to learn.
- Pre-assessment helps avoid boredom; students are more mentally engaged when the learning is directly relevant to their
needs and interesting to them.
- Pre-assessment saves instruction time when teachers do not re-teach what students already know.
- Pre-assessment enables teachers to form appropriate flexible groups— a key principle of a differentiated classroom.
- Pre-assessment supports the use of compacting and tiered instruction to match students’ readiness.
- Pre-assessment allows students to demonstrate and get credit for the concepts and skills previously mastered.
To make sure that all modules have the same structure, Module Developers should use a 20-question, multiple-choice, preassessment that briefly skims the most important content of the module. The answer key also needs to be provided, as well as
pedagogical comments for learners.
*One relevant image must be inserted here.
Title of Pre-assessment: Review of Basic Mathematics
Rationale: Basic mathematics is the pre-requisite for Number theory
QUESTIONS
1. Find the value of x in 2(2x + 2) = 64
a) 3
1
b) 5
c) 1
d) 2
Kingore, B. (2004). Differentiation: Simplified, Realistic and Effective. Pieces of Learning : Beavercreek, Ohio
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2. Solve the simultaneous
a) 7, 2
b) 1, 8
3x + 2y = 22
x + y =9
c) 4, 5
d) 6, 2
3. Solve the quadratic x2 – 3x – 10 = 0
a) -5, 2
b) 5, -2
c) -5, -2
d) 5, 2
4. Find the inverse of the function g(x) = 2x – 3
a) g-1(x) =
( x  3)
2
b) g-1(x) =
(2 x  3)
2
c) g-1(x) =
( x  3)
2
d) g-1(x) =
(2  3 x)
2
5. Find the greatest common divisor(gcd) of 986 and 289
a) 17
b) 58
6. Solve the equation
a) 2
c) 9
d) 3
1
3
6x

 2
x2 x2 x 4
b) 4
c) 2
d) 3
c) 11 + 2ί
d) 10ί  3 ί²
c) 84
d) 97
7. Work out (2 – ί)(4 + 3 ί)
a) 8 + ί
b) 13
6
8. Find

( 4i + 2)
i 1
a) 6
b) 26
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9. Evaluate 8C2
a) 20
b) 28
c) 16
d) 4
10. The third term of a geometric sequence is equal to 1 and the 5th term is 16.
Find the seventh term.
a) 4
11.
b) 128
b) 60
c) 110
d) 140
Given the equation y = x2 + 5x – 14, find the turning point.
a) -2, 7
13.
d) 4096
If s= ut + ½ at², evaluate s when u = –3, a = 10 and t = 5
a) 30
12.
c) 256
b) -7, 2
c) -2½, 14⅛
d) -2½, -20 1
4
When factorised 36j – 48 becomes:
a) 12(3j – 4)
b)
b)64
15. Solving the equation
a) -2
d)
8(4j – 6)
m
- 11 = - 2 is
8
14. The solution to
a) 56
12(24j – 36) c)9(4j – 7)
c) 72
d) 96
6(7+y)-2(5y-1)=12(3y+5)-16(y-5)
b) -4
c) -3
yields
d) 2
16. A 20 cm long straw is the longest that can fit into a cylindrical can with
radius of 6 cm. The height of the can in centimetres is closest to:
a) 8
17.
b) 15
c) 16
d) 9
Given the equation y = - x² + 2x + 8, find the x-intercepts.
a) - 2, 4
b) 2, - 4
Module Development Template
c) 2, 4
d) -2, - 4
8
18.
R
The value of angle ⊖ is
65 cm
67 cm
a)
0.570
b)
55.10
c)
430
d)
67.20
⊖
P
p
19.
For the sequence 7, 16, 25, 34, …. ……… the 56th term is
a) 495
20.
Q
73 cm
b) 640
c) 55
d) 502
Each interior angle of a regular polygon is 140 0 .How many sides does it
have
a) 5
b) 9
c) 11
d) 7
Title of pre-assessment: Review of Basic Mathematics
ANSWER KEY
1
d
11
c
2
c
12
d
3
b
13
a
4
a
14
c
5
a
15
b
6
a
16
c
7
c
17
a
8
d
18
b
9
b
19
d
10
c
20
b
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*One relevant image must be inserted here.
Title of Pre-assessment : Review of Basic Mathematics
PEDAGOGICAL COMMENT FOR LEARNERS
(100-200 words)
A learner’s entering behaviour determines whether he / she will have grasp of the
Number theory module. Number theory module builds on basic mathematics. Thus,
the pre-assessment is a gauge of how well you are conversant with basic
mathematics. It indicates your level of preparedness. Learners should revise basic
mathematics if they have problems in scoring the pre-assessment.
It is highly recommended that learners revise basic mathematics before and after the
pre-testing so as to improve on their perfomance in this module. Number theory is a
branch of abstract mathematics that utilises many mathematical notations. Before
starting the module, revise all the mathematical notations encountered from
secondary schools and in basic mathematics.
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10. KEY CONCEPTS (GLOSSARY)
Module Developer Writing Tip. This section contains short, concise definitions of terms used in the module. The key concept
section is often called glossary, it helps learners with terms with which they might not be familiar in the module. Learners
especially appreciate glossaries in an open and distance education context because there is no teacher in front of them, when
they encounter an unfamiliar word and cannot easily figure out its definition in the text.
Module Developers should select at least the 10 most important terms to include in the glossary. When writing the entries in
the glossary, the term needs to be in capital letters and boldface type. The definition must be brief, concise and clearly written.
For example :
ISOTOPE. An isotope is any of several different forms of an element each having different mass. Two isotopes of an element
will have nuclei with the same number of protons (the same atomic number) but different numbers of neutrons. Therefore,
isotopes have different mass numbers, which give the total number of nucleons - the number of protons plus neutrons. The
word isotope, from Greek meaning « at the same place » , comes from the fact that all isotopes of an element are located at
the same place on the periodic table. (source : Wikipedia, consulted August 9, 2006).
It is important to highlight that glossaries differ from dictionaries. Glossaries only provide the uses of the terms that are relevant
to the content of the module. Dictionaries provide more complete definitions.
1.
ALGORITHM: An algorithm is a procedure for solving a problem in a
finite number of steps
2.
INTEGER: Integers mean any element of the set { …-3, -2, -1, 0, 1,
2,3,…}
3.
PRIME NUMBERS: A prime number is a number that has only two divisors:
1 and itself.
4.
EVEN NUMBERS: Even numbers are numbers that can be divided by 2
without remainders.
5.
ODD NUMBERS: Odd numbers are numbers that have remainders when
divided by 2
6.
EUCLIDEAN ALGORITHM: Is a systematic procedure for finding the greatest
common divisor of two integers. Euclid was a Greek mathematician ( c. 400
B.C ) that developed the algorithm.
7.
A DIOPHANTINE EQUATION: Is a polynomial equation with integer
coefficients for which only integer solutions are allowed. e.g mx = k, where m
and k are integers and m  0, is a linear first degree Diophantine equation.
(Diophantine equations are named after the Greek mathematician Diophantus
of 3rd century A.D)
8.
LEMMA, THEOREM, COROLLARY: mean a “True Statement”
9.
A GAUSSIAN INTEGER: Is a complex number whose real and imaginary
part are both integers i.e a + bί where a and b are integers.
10.
THE NORM OF A GAUSSIAN INTEGER: Is the natural number defined as
N(a + bί) = a² + b²
11.
THE MODULUS OF A GAUSSIAN INTEGER: Is simply its complex modulus|
a + bί | =
12.
a2  b2
CONJUGATE: A conjugate of ( a + bί ) is ( a – bί)
11. COMPULSORY READINGS
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Module Developer Writing Tip. In open and distance education, learning often takes place through reading. In this section,
Module Developers should provide readings to the students. At least three relevant readings (approximately 5-10 pages
each) must be provided for each module. These readings should help learners understand the topics covered in the module.
For each reading, Module Developers need to provide the complete reference (APA style), as well as a 50 word abstract
written in a way that motivates the learner to read the text provided. The rationale for the reading provided should also be
explained (maximum length : 50-75 words). An electronic version of each reading is required.
Important note : the readings must be copyright free. That is, they must either be written by the Module Developer or be from
open access content. Open access (OA) is the free online availability of digital content (Wikipedia). Module Developers can
consult the Directory of Open Access Journals (http://www.doaj.org/) for readings that could be relevant. Modules that do not
comply with this will not be accepted.
*One relevant image must be inserted here.
Reading # 1: Wolfram MathWorld (visited 03.11.06)
Complete reference : http://mathworld.wolfram.com/NumberTheory.html
This reference gives the much needed reading material in Number
Theory. Learners are advised to critically check and follow the given the proofs of
Lemmas. In addition, the reference has a number of illustraions that empower the
learner through different approach methodology.
Abstract :
Rationale: The reference enables learners to analise number theory through the
abstract approaches that many learners fail to visualise. By reading through, the
learner will appreciate the technical inferences to Lemmas, Corollaries, theorems
and Propositions that are used in the various proofs.
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Reading # 2: Wikipedia (visited 03.11.06)
Complete reference : http://en.wikipedia.org/wiki/Number_Theory
Abstract : Wikipedia should be the learners closest source of reference in Number
Theory. It is a very powerful resource that all learners must refer to understand
abstract mathematics. Moreover, it enables the learner to access various arguments
that have puzzled mathematicians over the centuries.
Rationale: It gives definitions, explanations, and examples that learners cannot
access in other resources. The fact that wikipedia is frequently updated gives the
learner the latest approaches, abstract arguments, illustrations and refers to other
soucers to enable the learner acquire other proposed approaches in number theory.
Reading # 3: MacTutor History of Mathematics (visited 03.11.06)
Complete reference :
http://www-history.mcs.standrews.ac.uk/Indexes/Number_Theory.html
Abstract : MacTutor is a must read for interest and knowledge of the history of
Number Theory. It gives accounts of how theorems,propositions, corollaries and
lemmas have daunted mathematicians over the centuries. Fermat’s last theorem is
well illustrated as a very simple concept that a class / grade three pupil can
understand.However, the proof of the theorem dodged matheticians for over 300
years from the year 1637 to the year 1995.
Rationale: History of mathematics as approached in MacTutor not only gives the
historical aspects of number theory but also challenges learners to proof theorems,
propositions,lemmas, and corollaries that have not been proved. The learner appreciates
the challenges of proofs by many approaches such as induction and contradiction. Thus the
reference is suitable for a variety of mathematical approaches that every number theory
learner needs to know to enhance knowledge and consolidation of abstract mathematics.
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12. COMPULSORY RESOURCES
*Two relevant images must be inserted here.
Module Developer Writing Tip. In this section, Module Developers should provide at least two, copyright free, relevant,
compulsory resources other than a written text or a web site. These could be a video file, an audio file, a set of images, etc. For
each resource, Module Developers should provide the complete reference (APA style), as well as a 50 word abstract written in
a way that motivates the learner to use the resources provided. The rationale for the resource provided should also be
explained (maximum length : 50-75 words). An electronic version of each resource is required.
Important note : the resources must be copyright free. That is, they must either be created by the Module Developer or be
from open access content. Module Developers are encouraged to use open content learning objects from the following
ressources2 :
GEODE (http://www.uw-igs.org/search/) or "Global Education Online Depository and Exchange," is a repository of Global
Studies learning objects maintained by the Institute of World Affairs at the University of Wisconsin-Milwaukee. The edited
collection may be searched by country, region, file format, language, or keyword.
MERLOT (http://www.merlot.org/), short for the "Multimedia Educational Resource for Learning and Online Teaching," is a free
and open resource designed primarily for faculty and students of higher education. MERLOT includes links to online learning
materials along with annotations such as peer reviews and assignments. To learn more about the MERLOT project, visit
http://taste.merlot.org/. For the latest news from MERLOT, visit http://taste.merlot.org/portal/grapevine/. Finally, to browse the
subsets of the whole MERLOT collection that are focused on specific disciplines, visit http://www.merlot.org/home/Sites.po
Canada Learning Object Project eduSource (http://www.edusource.ca/). eduSource is a Canada-wide project to create the
infrastructure for a network of interoperable learning object repositories. A repository differs from standard web materials by
providing teachers, students and parents with information that is structured and organized to facilitate the finding and use of
learning materials regardless of their source location. The eduSource project is based on national and international standards;
it is bilingual (French/English) and it is accessible internationally.
Burrokeet (http://www.burrokeet.org/) is an Open Source Software tool that assists in the creation of Learning Objects from
existing content. It is able to take, as input, a wide range of document formats and export them as consistently styled content
within Learning Objects. This frees the content developer to focus on the quality of the content without having to overly concern
themselves with presentation. Similarly, editors of learning objects need not concern themselves with ensuring authors use the
same development tool, they are free to use whatever the tool with which they are most familiar. As a result Burrokeet
enhances the reusability of content within learning objects.
VLORN (http://www.flexiblelearning.net.au/vlorn/). The VET Learning Object Repository Network (VLORN) is a network of
organizations in the Australian vocational education and training (VET) that contribute via agreed standards to enable the
discovery and use of learning objects. VLORN was established in 2004 through Australian Flexible Learning Framework
funding. See http://www.flexiblelearning.net.au/ for more information on the Framework.
[edit]
LON-CAPA (http://www.loncapa.org/) is a distributed network with participants from currently over 40 colleges and universities,
as well as 40 K-12 schools (mostly in the US), who share a common pool of approximately 150,000 reusable learning objects.
Module Developers are also encouraged to visit the following web site : Collections of learning objects
(http://www.uwm.edu/Dept/CIE/AOP/LO_collections.html). Modules that do not comply with this will not be accepted.
2
Adapted from Wikipedia.
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Resource #1
Maxima.
Complete reference : Copy of Maxima on a disc is accompanying this course
Abstract : The distance learners are occasionally confronted by difficult mathematics
without resources to handle them. The absence of face to face daily lessons with
teachers means that learners can become totally handicapped if not well equipped
with resources to solve their mathematical problems. This handicap is solved by use
of accompanying resource: Maxima.
Rationale: Maxima is an open-source software that can enable learners to solve
linear and quadratic equations, simultaneous equations, integration and
differentiation, perform algebraic manipulations: factorisation, simplification,
expansion, etc This resource is compulsory for learners taking distance learning as
it enables them learn faster using the ICT skills already learnt.
Resource #2 Graph
Complete reference : Copy of Graph on a disc is accompanying this course
Abstract : It is difficult to draw graphs of functions, especially complicated functions,
most especially functions in 3 dimensions. The learners, being distance learners, will
inevitably encounter situations that will need mathematical graphing. This course is
accompanied by a software called Graph to help learners in graphing. Learners
however need to familiarise with the Graph software to be able to use it.
Rationale: Graph is an open-source dynamic graphing software that learners can
access on the given CD. It helps all mathematics learners to graph what would
otherwise be a nightmare for them. It is simple to use once a learner invests time to
learn how to use it. Learners should take advantage of the Graph sofware because it
can assist the learners in graphing in other subjects during the course and after.
Learners will find it extremely useful when teaching mathematics at secondary
school level.
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13. USEFUL LINKS
*One relevant image must be inserted here.
Module Developer Writing Tip. In open and distance education, learning often takes place through reading information
presented on a web site. In this section, Module Developers must provide a list of at least 10 relevant web sites. These useful
links should help students understand the topics covered in the module. For each link, Module Developers should provide the
complete reference (Title of the site, URL), as well as a 50 word description written in a way to motivate the learner to read the
text provided. The rationale for the link provided should also be explained (maximum length : 50 words). A screen capture of
each useful link is required. Note that Toby Harper, Dr. Thierry Karsenti’s assistant, will help all participants with that aspect.
Important note : the links listed should be as stable and come from official sources, as much as possible.
Useful Link #1
Title : Fermat’s Last Theorem
URL : http:www. http://www-groups.dcs.st-and.ac.uk~history/HistTopics/Fermat’s
last theorem.html
Screen capture :
Description: Fermat’s Last Theorem is simply stated as: x3 + y3 = z3 . Despite this
simplicity, it’s proof dodged mathematicians for over three hundred years from 1637
to 1995. Why does such a simple cubic equation that can be understood by Primary
school children dodge mathematicians? Find out from the website.
Rationale: Pythagorean triples have been used from the ancient times by the babylonians
and Egyptians in the construction industry. Pythagoras, a mathematician, documented the
theory behind the pythagorean triples. He however did not try the cubic equations and this
became the subject that dodged mathematicians after Fermat’s proposition. The history
behind this theory is very facinating to the mathematics mind and is a must read for Number
Theory learners.
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Useful Link #2
Title : Wikipedia
URL : http:www. http://en.wikipedia.org/wiki/Number_Theory
Screen capture :
Description: Wikipedia is every mathematician’s dictionary. It is an open-resource that is
frequently updated. Most number theory students will encounter problems of reference
materials from time to time. Most of the number theory books available cover only parts or
sections of number theiry module. This shortage of reference materials can be overcome
through the use of Wikipedia. It’s easy to access through “Google search”
Rationale: The availabilty of Wikipedia solves the problem of crucial learning materials in all
branches of mathematics. Learners should have first hand experience of Wekipedia to help
them in their learning. It is a very useful free resource that not only solves student’s
problems of referece materials but also directs learners to other related useful websites by
clicking on given icons. It’s usefulness is unparalleled.
Useful Link #3
Title : Mathsguru
URL :
http://www.bbc.co.uk/education/asguru/maths/13pure/01proof/01proof/05induction/in
dex.shtml
Screen capture :
Description: Mathsguru is a website that helps learners to understand various branches of
number theory module. It is easy to access through Google search and provides very
detailed information on verious number theory questions. It offers explanations and
exapmles that learners can understand easily.
Rationale: Mathsguru gives alternative ways of accessing other subject related topics,hints
and solutions that can be quite handy to learners who encounter frustrations of getting
relevant books that help solve learners problems in number theory. It gives a helpful
approach in computation of number theory by looking at the various branches of the number
theory module.
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Useful Link #4
Title : Mathworld Wolfram
URL : http:www. http://mathworld.wolfram.com/NumberTheory.html
Screen capture :
Description: Mathworld Wolfram is a distinctive website full of number theory solutions. It
gives both the corollaries, Lemmas, and propositions and there proofs. Learners shouls
access this website quite easily through Google search for easy reference. Wolfram also
leads learners to other useful websites that cover the the same topic to enhance the
understnding of the learners.
Rationale:Wolfram is a useful site that provides insights in number theory while providing
new challenges and methodology in number theory. The site comes handy in mathematics
modelling and is highly recommended for learners who wish to study number theory and
other branches of mathematics. It gives aid in linking other webs thereby furnishing learners
with a vast amount of information that they need to comprehend in number theory.
Useful Link #5
Title : Proof of Fermat’s last Theorem by Wiles
URL : http:www. http://www.pbs.org/wgbh/nova/proof/wiles.html
Screen capture::
Description: Fermat’s last theorem dodged mathematicans for over three hundred years
from 1637 to 1995. In 1995, Professor Wiles proved Fermat’s Last Theorem. It is an
interesting account of this taxing exercise of proving Fermat’s last theorem that many
mathematicians all over the world had tried without success. Wiles first encountered the
problem in primary school and attempted severally to prove it without success. In 1995,
Wiles, then a professor of Mathematics, proved his childhood dream: Fermat’s last theorem.
Rationale: The website highlights the intrigues of proving mathematical propositions and
opens up other untouched propositions that every learner can attempt and contribute to the
ever challenging world of mathematics. It is important for learners to know that “not all
mathematics “ have been discovered and they have room to improve by proving some
propositions that have been pending for many years.
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14. LEARNING ACTIVITIES
Module Developer Writing Tip. The learning activities are at the heart of each module. They represent the tasks or activities
learners must accomplish in order to meet the module’s objectives.
Module Developers should write at least four learning activities per module. For each learning activity, clear learner
guidance needs to be provided (a section called “INSTRUCTIONS”), including the resources required in order to accomplish
the learning tasks.
Module Developers should write a summary (maximum length : 100 words) for each learning activity. Each summary should
be followed by a series of 5 key words.
Module Developers are also required to provided a detailed description for each learning activity (maximum length : 250
words).
As students learn best when they are actively involved in the process, the learning activities developed should focus on active
learner involvement. Also, as many studies report that, regardless of the subject matter, students working collaboratively in
small groups tend to be more motivated, satisfied and to learn more than when the same content is presented in other
instructional formats, one of the four learning activities must include a collaborative learning component. Module Developers
who need more information on collaborative learning can consult Annex 4 (What is collaborative learning).
Module Developers should find a variety of learning activities (practice activities, exercises, assignments, projects, reports,
presentations, problem solving activities, etc.) which will suit a variety of learning styles, within the African context.
Module Developers are required to make sure that information and communication technologies (ITC) are integrated in
most learning activities. When specific tools are used in one activity, these need to be mentioned. These also have to be open
source tools (such as Open Office tools : www.openoffice.org).
Module Developers should assure that the readings, the resources and the useful links mentioned in the sections above
are all integrated in the learning activities. Because of the reality of the African context, some learning activities and learner
support must be provided for students who have limited access to technology.
Also, Module Developers have to make sure that staff feedback loops are integrated into the Module.
Each learning activity must include at least one formative evaluation component. A formative evaluation is an ongoing
assessment throughout the learning process. It is essential to help the instructor understand the learners’ mastery of the
learning objectives of the module. Module Developers should understand that formative evaluation must also offer guidance on
how the learner’s work or performance can be improved.
Module Developers must also address the type of feedback required by the instructor of the course. This feedback must go
beyond managerial functions and focus on the learning process. In fact, it should be recommended to instructors that feedback
focus on the performance, with advice on what could be done to further improve that performance. Research clearly shows that
feedback improves learning when it gives students specific guidance both on the strengths and weaknesses of the students.
Formative assessments should take many different forms in the module. For example, some learning activities could integrate
on-line small-group discussions among students (these need to be highly structured), whole class discussions on the module
electronic discussion forum (again, this needs to be highly structured), journal entries, portfolios entries, surveys, analytical
observations, problem solving activities, reading activities (followed by questions), etc. The Module Developer needs to
indicate if this formative evaluation counts for the final mark of the module, as well as its value. An answer key also needs to be
provided for each formative evaluation. Also, Module Developers must indicate how learners will submit their answers to the
instructor. Will the summary evaluation be emailed to the instructor ? Will it be submitted online ? Will there be an online test ?
Will there be an on-site evaluation ? Finally, formative evaluations should include opportunities for students to respond to the
module to provide feedback to the instructor.
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Learning Activities
Title of Learning Activity
Unit 1 (60 Hours): Properties of Integers and Linear Diophantine
Equations
Summaryof Unit 1( Multiple Activities)
The approach used in this module involves reviewing ordinary day today life
situations and linking up to the theory of numbers. Due to the abstract nature of
number theory, reference to lemmas, theorems, and proofs is given to learners as
reference materials. Learners may be involved in writing of simple computer
programs to carry out calculations based on a mathematical skill and formula.
Learners need not be necessarily computer literate to carry out the activities.
Theorems have been quoted but the proofs are given for group work activities with
the reference materials. It must be noted that this module is concerned with abstract
mathematics and activities of this module are centered around the computational
number theory, theorems, lemmas, propositions, corollaries and their proofs .
Readings
All of the readings for the module come from Open Source text books. This means
that the authors have made them available for any to use them without charge. We
have provided complete copies of these texts on the CD accompanying this course.
You will be referred to specific sections of these books when they are needed in this
study guide.
1. Elementary Number Theory, By W. Edwin Clark, University of South Florida,
2003. (File name on CD: Elem_number_theory_Clarke)
2. Elementary Number Theory, By William Stein, Harvard University, 2005 (File
name on CD: Number_Theory_Stein)
3. MIT Open Courseware, Theory of Numbers, Spring 2003, Prof. Martin Olsson
(Folder name on CD: MIT_Theory_of_Numbers)
Internet Resources
These are general resources that apply to the whole module. They offer
opportunities for additional reading and for background study. Specific internet
resources will be detailed at the appropriate section in this study guide.
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COURSE PREREQUISITES
This module assumes learners are familiar with numbers covered in basic
mathematics. The following properties of numbers are assumed:
1) Commutative laws: p + q = q + r and pq = qp
2) Associative laws: p + (q + r) = (p + q) + r = p + q + r
3) Laws of Indices:
a) am  an = am + n
b) am  an = am - n
c) (am)n = amn where m and n  0.
d)
1
 a-m
am
e) a0 =1, a  0
f) a-n =
g)
1
an
1
n
a  n a ,n  0.
h) a a
m
n
 n a m ,n  0.
i) am  bm = (ab)m
j)
am
a
= 
m
b
b
m
p  p if p is positive and p   p if p is negative.
The modulus function gives the numerical value of an input. It converts negative
numbers to positive and is written as y  x and read as “y equals mod x”
Example:
Stat the value of |7  x| for x = 15.
Solution: When x =15, |7 15| = |8| = 8
4) Absolute or Modulus value of
5) Solutions of quadratic equations: you should be able to solve linear and quadratic
simultaneous equations using an algebraic method.
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FORMATIVE EVALUATION 1:
Exercise: Indices
Solve for x:
1. 4x+2 = 82x
2. 22x+1  5(2x) +2 = 0
3. logx6 = ½
Evaluate:
3
 4 2
4.  
 49 
5. log10 0.001
Answers
1. x=1
2. x=1, x = 1
3. x=36
3
7
4.  
 2
5. x= 3
Notation
1) If p is divisible by q, we write p  q. If p is not divisible by q, we write p
2)  means “for all”
3)  means “such that”
4) Iff means “ if and only if”
5)  means “is a member of”
6) Z means “set of integers”
7)  means “ implies “
8)  means “there exists”
9) ≡ means “ equivalent”
10)  means “ is not a member of“
q
Let p,q and r be integers. Then:
a)
b)
c)
d)
e)
p|q, a>0,q>0  p  q
p|q  p|qr ,  integers r
p|q, p|r  p|( qx + ry) for x,y  Z
p|q, q|p  p =  q
p|q , q|r  p|r
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MATHEMATICAL PROOFS: BY INDUCTION AND CONTRADICTION
The Number Theory module uses mathematical induction and indirect proof or proof
by contradiction extensively.
Example 1: Proof by Induction
Prove by mathematical induction that
1 + 2 + ……………. + m =
m( m  1)
2
Proof:
Step 1: Technique of Induction
Mathematical induction proves by checking if a proposition hold’s for m=1, and m=
k+1 whenever it holds for m=k, then the proposition holds for all positive integers m=
1,2, 3,……..
Step 2: Substitute m=1 in the equation:
1=
1(1  1)
=1
2
Step 3: Assume that the formula holds for m= k
1 + 2 + ……………. + k =
k ( k  1)
2
Step 4: Proof that the formula holds for m = k+1.
1 + 2 + ……………. + k + ( k + 1) =
(k  1){( k  1)  1}
2
We write:
1 + 2 + ……………. + k =

k ( k  1)
2
k ( k  1)
(k  1){( k  1)  1}
k (k  1)  2k  2
(k  1){( k  2)
+ ( k + 1) =

=
2
2
2
2
(k  1){( k  2)
(k  1){( k  2)
=
{ by factorization} [ PROVED ]
2
2
This is proof by inductionExample: Proof by Induction

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Example 2:
Prove by induction that for every positive integer n, then:
1² +2² + 3² + 4² +…………. + n² =
n(n  1)( 2n  1)
6
Step 1: Technique of Induction
Mathematical induction proves by checking if a proposition hold’s for n = 1, and
n = k + 1 whenever it holds for n = k, then the proposition holds for all positive
integers n= 1,2, 3,……..
Step 2: Substitute n = 1 in the equation:
1=
1{1  1}{( 2  1)  1}
=1
6
Step 3: Assume that the formula holds for k
k (k  1)( 2k  1)
6
Step 4: Proof that the formula holds for n= k+1.
1² +2² + 3² + 4² +…………. + k² =
1² +2² + 3² + 4² +…………. + ( k + 1)² =
(k  1)( k  1  1){2(k  1)  1}
6
We write
1² +2² + 3² + 4² +…………. + (k + 1)² = (1² + 2² + 3² + 4² +…………. + k²) + (k + 1)²
k (k  1)( 2k  1)

+ (k + 1)²
6

(k  1)( 2k 2  k  6k  6)
6

(k  1)( k  2)( 2k  3)
{by factorization} [ PROVED ]
6
This is proof by induction
READ: Proof by Induction
1. Elementary Number Theory,
By W. Edwin Clark, 2003, pages 2 -7
Internet Resource
http://www.bbc.co.uk/education/asguru/maths/13pure/01proof/05induction/index.sht
ml
Read the two pages about proof by induction at this web site.
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FORMATIVE EVALUATION 2:
Exercise: Proof by Induction
1. Prove that 1 + 2 + 2² +……. + 2n = 2n+1 – 1 for n  1.
2. Prove that 1 + 3 + 5 + 7 +…….. + (2n – 1) = n2
a + ar + ar2 +…+ arn =
3. Prove that
4. Prove that
14
+
24
+
34
+
44
+ ..…+
a(1  r n 1 )
, for n > 0
1 r
n4
n(n  1)( 2n  1)(3n 2  3n  1)
=
30
5. Prove that for n < 2n for all positive integers n.
6. Prove that (ab)n = anbn
7. Prove that 1 + 4 + 7 + 10 + …… + (3n  2) =
n n
( 3  1)
2
Odd and Even Numbers
Even and Odd Numbers Activity
Case 1
What do you understand by even and odd numbers?
Give one simple way of distinguishing even from odd numbers?
How many months of the year have number of days that are odd?
How many even years are there from 1960 to 2010 ?
Answer
Numbers divisible by 2 are called Even and numbers not divisible by 2 are called
Odd.
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Flow Chart for Testing Even and Odd numbers.
Start
READ in an
integer N
Calculate M =( N / 2)  2,
ignoring Ignoring the
remainder
yes
no
?
M=N
WRITE N
“IS ODD”
WRITE N
“IS EVEN”
STOP
Procedure:
1.
2.
3.
Input an integer N
Calculate M as indicated
Make the decision if N is even or odd.
The activity is a flow chart representation of sorting even and odd integers.
Tabulate your results:
Number, N
Even
Odd
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FORMATIVE EVALUATION 3:
Exercise
Modify the flow chart to test whether a number is divisible by 3.
Answer
Change the statement M=N/2*2 to M=N/3*3.Also change the FORMAT statements
to write appropriate messages.
Reflection
1. As a teacher trainee, how would you teach indices and integers to reflect
real life situations? Think of practical approaches to the teaching of integers
that involve a learner’s day to day life experiences.
2. Number lines have been used to teach computation of positive and negative
integers. How can a teacher use the number line without missing out the
real meaning
basic operations of division, addition, subtraction and
PROPERTIES
OFofINTEGERS
multiplication as relates to real life? For example:  2 ×  2 = 4
DIVISOR
A divisor of an integer n, also called a factor of n, is an integer which evenly divides
n without leaving a remainder.
Example:
7 is a divisor of 35 because 35|7 = 5. We also say 35 is divisible by 7 or 35 is a
multiple of 7 or seven divides 35 and we usually write 7|35.
In general, we say m|n (read: m divides n) for non-zero integers. If there exists an
integer k such that n=km. Thus divisors can be negative as well as positive e.g.
divisors of 6 are 1,2,3,6,-1,-2,- 3,-6 but one would usually mention the positive ones
1,2,3 & 6. 1 and –1 divide (are divisors of) every integer, every integer is a divisor of
itself and every integer is a divisor of 0.
A divisor of n that is not 1,-1, n or –n is known as non-trivial divisor, numbers with
non-trivial divisors are known as composite numbers while prime numbers have nontrivial divisors.
If a|b=c, then a is the dividend, b the divisor and c the quotient.
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The Remainder for natural Numbers.
If a and d are natural numbers, with d non-zero, it can be proved that there exist
unique integers q and r, such that a = qd + r and 0  r < d. The number q is called
the quotient, while r is called the remainder.
Example:
1) When diving 17 by 10, 1 is the quotient and 7 is the remainder because 17 =
1  10 + 7
2) 22 / 4 = 5  4 + 2
5 is the quotient & 2 the remainder.
3) When dividing 42 by 7, 6 is the quotient and 0 is the remainder, because
42 = 7  6 + 0
The Case of General Integers.
If a and d are integers, with d non-zero, then a remainder is an integer r such that
a=qd+r for some integer q, and 0<= |r| <= |d|
When defined this way, there are two possible remainders.
Example:
The division – 37 by 5 can be expressed as either – 37 = 8  (-5) +3 or
– 37 = 7  (-5)+(-2). So the remainder is then 3 or –2
Note: When dividing by d, if the positive remainder is r1, and the negative one is r2 ,
then r1 = r2 + d
Modulo Operation.
The modulo operation finds the remainder of division of one number by another.
Given two numbers a and n, a modulo n { abbreviated a mod n } is the remainder, on
division of a by n. eg 10 mod 3 evaluates to 1 and 12 mod 3 evaluates to 0 i.e 1 and
3 are the remainders after division.
Divisibility
Definition: An integer p is divisible by an integer q iff  an integer r  p = q  r
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The Division Theorem
If m and n are integers, with n  0, then there exists unique integers q and r,
0 ≤ r I n I,  m = qn + r.
The integers
a) m is called the dividend
b) q is called the quotient
c) n is called the divisor
d) r is called the remainder
Case 1
Dividing an 11 hectares piece of land among 5 people. What does each get?
Each person gets a whole number and a fraction.
In this case, identify the dividend (a), quotient (q), divisor (b) and remainder(r ).
Examples:
If m and n are integers, with n
and r,
0 ≤ r I n I,  m = qn + r.
 0, then there exists unique integers q
n
q
r
m= qn + r
The integers
2
7
1
m=7(2) + 1
m is called the dividend
5
6
3
m =5(6) + 3
q is called the quotient
- 10
2
1
m = -10(2) + 1
-9
5
8
m = -9 (5) +8
n is called the divisor
FORMATIVE EVALUATION 4:
TASK : Factors
1.Fill in the table
m
n
7
2
7
-3
-7
3
-7
-3
Module Development Template
q
r
Solution
29
Definition:
A natural number which divides into another an exact number of times is called a
factor.
Examples:
 Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
 Factors of 15 are 1, 3, 6,and 15.
FORMATIVE EVALUATION 5:
Exercise: Factors
What are the factors of:
1. 20
2. 28
3. 36
4. 120
5. 169
6. 180
Common Multiples
An integer that is divisible by two integers p and q is called a multiple of p and q. The
common multiples of 2 and 3 are 0, 6, 12, 18, 24,…. And the common multiples of
4,and 5 are 0, 20, 40,60,…….
FORMATIVE EVALUATION 6:
Exercise: Common Multiples
List the first 8 multiples each of:
1). 3
2). 7
3). 11
4). 23
5). 61
6). 138
Least Common Multiple (LCM )
Market Story
Madam Safia goes for shopping in the nearest shopping center to
her home where things are measured in tins. She visits three
shops, which use different sizes of tins. Shop A uses 2 litre tins,
shop B uses 4 litre tins and shop C uses 5 litre tins. She needs to
carry a container that enables her buy a whole number of tins
irrespective of which shop she visits. What is the minimum
volume of the container she needs to carry?
Definition:
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The least common multiple of p and q is defined as the smallest positive integer that
is divisible by both p and q. It may be denoted as [p, q].
Examples:
 [4, 9] = 36
 [3, 4] = 12
 [7, 8] = 56
Calculation of LCM using prime factors
Example: Find the LCM of 16, 24 and 840.
STEP 1: Express each of the numbers as prime factors
16 = 24
24 = 23  3
840= 23  3  5  7
STEP 2: Pick out the highest power of each of the prime factors that appears. The
factors need not be common. For example, the highest powers of 2,3 ,5 and 7 are
4,1, 1, and the LCD becomes 24357=1680.
Exercise: Finding the LCM
Find the LCM’S of
1. 18, 20,and 24
2. 30, 45,and 50
3. 252, 990 and 3150
4. 450, 2100 and 990
Common Divisors
Definition:
An integer p is a common divisor of q and r if p|q and p|r.
Greatest Common Divisor
Given three numbers 20,24, and 28, what is the greatest number that can divide
each of this numbers? How do you calculate this number ?
Definition:
Any two integers p and q have at least one positive divisor in common, called
greatest common divisor ( gcd). If at least one of the integers p and q is different
from zero, then there exists a largest positive integer d which divides both p and q.
This integer d is called the greatest common divisor ( gcd) of p and q and may be
denoted as gcd (p,q) or (p,q).
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Examples:
 gcd(6,12) = 3
 gcd(0,18) = (0,18)=18
 gcd(9, 27) = 9
 gcd(14, 28) = 7
Calculation of gcd using prime factors
Example:
Find the gcd of 60, 100, and 840.
STEP 1:
Express each of the numbers as prime factors
60 = 2²  3  5
100= 2²  5²
840= 23  3  5  7
STEP 2:
Pick out the highest common power of each common factor. The product of these
highest powers gives the gcd.
For example, the common prime factors are 2 and 5. The highest powers of 2 and 5
which are common are 2²  51=20 their gcd.
FORMATIVE EVALUATION 7:
Exercise: Finding the gcd
Find the gcd’s of
1. 540,72,and 378
2. 105,546,and 231
3. 1125 and 675
Readings:
1. Elementary Number Theory, by Stein, October 2005,
pages 5 -7
2. Greatest Common Divisor MIT: Units 1 & 2 Notes,
pages 1 & 2 each
3. Elementary Number Theory, By W. Edwin Clark, page
10 -14
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FORMATIVE EVALUATION 8:
Exercise: Prove the following corollaries:
1. For every m > 0, m(b,c)= (mb,mc)
1
a b
2. If d|a, d|b, d> 0,then  ,  = (a, b)
d
d d
Proof the following propositions
1. If (a,m)= (b,m)= 1 then (ab,m)=1
2. If c|ab and (b,c) =1 then c|a
Reference: MIT Notes 7 Feb 2003 ( Common Divisor) page1 & 2
Reflection
1. Think of suitable examples that can be used in the teaching of gcd’s and
lcm’s that learners can quickly identify with as part of their daily life.
2. Good teaching helps learners’ to integrate the theory with the practical. How
can a teacher integrate the gcd and lcm into the “domestic mathematics” of
EUCLIDEAN
measuringALGORITHM
in their homes?
Eulidean algorithm (Euclid’s algorithm) is an algorithm to determine the greatest
common divisor (GCD or gcd ) of two integers by repeatedly dividing the two
numbers and the remainder in turns.
Description of the algorithm
Given two natural numbers m and n, check if n = 0. If yes, m is the gcd. If not, repeat
the process using n and the remainder after integer division of m and n {written as
m modulo n}
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Theorem: Euclidean algorithm
Either m is a multiple of n, or there is a positive integer k, and integers
q1,q2,……,qk, r1,r2,…….r k-1 ( and r = 0) such that
(0 ≤ r1 I n I )
m= q1 n + r1
(0 ≤ r1  r2 )
n= q2 r1 + r2
……
(0 ≤ rk-1 rk-2 )
rk-3= qk-1 rk-2 + rk-1
(0 ≤ rk )
rk-2 =qkrk-1
Example: Compute the gcd of 1071 and 1029.
Euclid ( 400 B.C) developed a systematic procedure for finding the greatest
common divisor of two integers. It is called the Euclidean algorithm.
a
b
Expression
Explanation
1071
1029
1071
1029
1071=10291+42
Step 2: The remainder of 1071 1029
is 42, which is put on the right, and the
divisor 1029 is put on the left.
1029
42
1029=4224+21
Step 3: Repeat step, dividing 1029 by
42, and get 21 as remainder.
42
21
42=212+0
Step 4: Repeat step 2again, since 42
is divisible by 21, we get 0 as
remainder, and algorithm terminates.
21
0
Step 1: Put the large number on the
left and the smaller one on the right.
Module Development Template
The number 21 is the gcd as required
34
Example: Compute the gcd of 1071 and 1029.
Examples: Illustration of Euclid’s algorithm in computing gcd
(Division algorithm)
Example 1
Example 2
Find the gcd of 5775 and 1008
Find the gcd of 2261 and 1275
Solution.
Solution.
m=5775 and n = 1008.
m=2261 and n= 1275
5775 = 5 x 1008 + 735
2261 = 1 x1275 + 986
1008 = 1 x 735 + 273
1275= 1 x986 + 289
735 = 2 x 273 + 189
986=3 x 289 +119
189 = 2 x 84 + 21
289=2 x119+51
84
119=2 x 51+17
= 4 x 21
Thus the gcd = 21, ie the largest Thus the gcd = 17.
integer that divides 5775 and 1008.
FORMATIVE EVALUATION 9:
Exercise: Finding gcd’s Using Euclidean Algorithm
Find the greatest common divisor in each case using Euclidean algorithm:
1. ( 276, 336, 396, 468, 972 )
2. ( 1387, 1292,722,836)
3. (924, 798, 1358,1827)
4. (60,84)
5. ( 190,72)
Solutions:
1). 12
2). 19
3). 7
4). 12
5). 2
FORMATIVE EVALUATION 10:
Exercise
Task:Use Elementary Number Theory by William Stein)
Attempt question 2.1, exercise 2.6 on page 38
Reading:
1. Elementary Number Theorem, by Stein, October 2005, page
8 -10
2. Euclidean Algorithm & Common Multiples MIT Unit 3, pages
1&2
1. Elementary Number Theory, By W. Edwin Clark, page 1 -33
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Reflection
Euclid’s algorithm is basically the familiar long division. Referring to the two
examples of Euclid’s algorithm, attach computational meaning to the various
steps of the long division. When we attach meaning to the division steps, does it
make more sense to the learners? Give appropriate wording to the division
steps to enable a colleague understand the algorithm.
Prime Numbers and Their Distribution
Introduction
The set of natural numbers is Z = {1,2,3,4,…}and
the set of integers N = {……-2,-1,0,1,2…..}
Definition: Prime and Composite
An integer p > 1 is prime iff it has no divisors d with 1< d< p. In other words, the only
positive divisors of p are 1 and p. We call p composite if p is not prime.
The number 1 is neither prime nor composite. The first few primes of N are
2,3,5,7,11,13,17,23,29,31,37,42,43,47,…… and the first few composite are
4,6,8,9,10,12,14,16,18,20,21,22,24,26,27,….
Definition:
Two integers p and q are relatively prime if gcd ( p,q) = 1.
Theorem
If p is composite, then p has a prime factor
Example:
Composite number 12 can be factored into primes i.e 12 =2 x 2 x 3, and
90 = 2 x 3 x 3 x 5
Fundamental Theorem of Arithmetic
Every integer greater than 1 is either prime or can be expressed as a product of
primes.
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Corollary:
The following are equivalent:
1. a and b have no common divisors i.e ( n|a and n|b)  n =  1.
2. (a,b)=1 i.e the subgroup generated by a and b is all of Z.
3.  m,n Z with ma +nb = 1.
Definition:
If any one of these three conditions is satisfied, we say that a and b are
Relatively Prime.
Theorem:
If a and b are relatively prime with a not zero, then a|bc = a|c
Proof:
Suppose a and b are relatively prime, c  Z and a|bc, then there exist m,n with
ma+nb =1, and thus mac + nbc = c. Now a|mac and a|nbc. Thus a|(mac + nbc) and
so a|c.
Theorem
Suppose p is a prime
1. If a is an integer which is not a multiple of p, then (p,a) = 1. In other words, if a
is any integer (p,a)=p or (p,a)=1.
2. If p|ab then p|a or p|b.
3. If p|a1,a2,…. an then p divides ai . Thus if each ai is a prime, then p is equal to
some ai.
The Unique Factorisation Theorem
Suppose a is an integer which is not 0,1 or -1. Then a may be factored into the
product of primes and except for order, this factorisation is unique. That is,  a
unique collection of distinct primes p1,p2,…….. , pk and positive integers s1,s2,….,sk
such that a= p 1S1 ,p 2S 2 ,….p kS k .
{E.H Connell, 2004}
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Group Work
1. Study the proof of fundamental theory of arithmetic.
Make sure you can proof this theorem during
examinations
Refer: Euclid’s proof of infinitely many primes in
Elementary Number Theory, By Stein,2005, pages
13 & 14
2. What is the largest known prime number?
3. Illustrate the primes of the form
a. ax + b
b. 4x  1
4. State the prime number theorem
Refer: Elementary Number Theory, By William
Stein,2005, page 15 & 18.
Solving Linear Diophantine Equations
Definition
A Diophantine Equation is a polynomial equation ( e.g mx = k, mx + ny = k ,etc)
with integer coefficients ( m and n ) for which only integer solutions are allowed.
First Degree Linear Diophantine equation
This is an equation in one variable for example, mx = k, where m and k are integers
and m  0, is a linear first degree Diophantine equation.
The linear Diophantine equation has an integer solution, x = k/m.
Diophantine Equations in Two Variables
These are of the nature mx  ny  k . ( m,n and k integers and m  0,n  0 )
This equation is solvable if k is the gcd (m,n), where m  0,n  0
Theorems
1. Given the integers m  0 and n  0, there exist integers x and y such that the
Diophantine equation mx + ny = gcd (m,n)
2. The Diophantine equation mx+ny = k, is solvable in integers iff gcd (m,n)
divides k.
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Activity: Solving Diophantine Equations
Example 1:
Solve the Diophantine equation
2772x + 390y = (2772,390)
Follow
through
the example
Solution:
STEP 1: Applying the Euclidean algorithm to find gcd of 2772 and 390
 2772 = 7 × 390 + 42…………………………………….( i)
 390 = 9 × 42 + 12……………………………………….( ii )
 42 = 3 × 12 + 6………………………………………….( iii)
The gcd = 6
STEP 2: Substitute the gcd in the equation ie 2772x + 390y = 6
Substitute backwards in ( iii), then (ii) and finally in (i) to obtain solutions for the
Diophantine
 6 = 42 – 3 × 12
 = 42 – 3 × ( 390 – 9 × 42) = 42  3(390)
 = 42 + 27(42)  3(390)
 = 28(42)  3(390)
 = 28(2772 – 7 × 390) – 3(390)
 = 28 (2772) – 196(390) – 3(390)
 = 28(2772) – 199(390)
i.e. ( mx + ny)
 x = 28, y = 199
Example 2:
Solve the Diophantine equation
7472x + 2624y = (7472, 2624)
STEP 1: Applying the Euclidean algorithm to find gcd of 7472 and 2624
 7472 = 3 × 2624 + 80…………………………………( i)
 2624 = 30 × 80 + 64……………………………………( ii )
 80 = 1 × 64 + 16…………………………………….…. ( iii)
The gcd = 16
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STEP 2: Substitute the gcd in the equation ie 7472x + 2624y = 16
Substitute backwards in ( iii), then (ii) and finally in (i) to obtain solutions for the
Diophantine
 16 = 80 – 1 × 64

= 80 – 1 (2624 – 30 × 80)

= 80  1(2624) + 30 × 80

= (1)80 + 30(80) – 1(2624)

= 31(80) – 1(2624)

= 31(7472 – 3 × 2624) – 1(2624)

= 31(7472) – 93(2624) – 1(2624)

= 31(7472) – 94(2624)

i.e. (mx + ny)

x= 31, y = 94.
Example 3:
Solve the Diophantine equation
803x + 154y = (803,154)
STEP 1: Applying the Euclidean algorithm to find gcd of 803 and 154
 803 = 5 × 154 + 33…….……………………………
 154 = 4 × 33 + 22……………………………………
 33 = 1 × 22 + 11..……………………………….….
(i)
(ii)
(iii)
The gcd = 11
STEP 2: Substitute the gcd in the equation ie 803x + 154y = 11
Substitute backwards in ( iii), then (ii) and finally in (i) to obtain solutions for the
Diophantine
 11 = 33 – 1 × 22

= 33 – 1(154 – 4 × 33)

= 33 154 + 4(33)

= 5(33) – 154

= 5(803  5(154)) – 154

= 5(803) – 25(154) – 154

= 5(803)  26(154)
 5(803)  26(154) ≡ 803x + 154y

x = 5 and y = 26
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FORMATIVE EVALUATION 11:
TASK : SOLVING DIOPHANTINE EQUATIONS
EXERCISE
Solve the Diophantine equations:
1.
2.
3.
4.
mx + ny = ( m,n) for m =5775, n= 1008.
18203x – 9077y = 17
32x + 14y = 22
35x + 61y =1
Solutions:
1. x = 11, y =-63
2. x = 17 x 742 = 12,597 y = 17 x 1486 = 25,262
3. Several solutions: x = -33, y = 77. The general solution is x= -33 +7i,
16i ( i =0, 1, 2, 3, 4,……)
4. x=7, y= -4
y = 77 –
(Kirch, 1974 & Clarke )
FORMATIVE EVALUATION 12:
Group Work: Solving Diophantine Equations
Find all the solutions x, y of each of the following
Diophantine equations:
1. 64x +108y = 4
2. 64x + 108y = 2
3. 64x + 108y =12
Reflection
1. In your own view, what are the essential steps in solving Diophantine
equations? How best can a teacher approach the solving of Diophantine
equations?
2. Identify the main areas that a teacher may need to emphasise when
teaching the solving of Diophantine equations.
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CONGRUENCES and INTEGERS ( MOD N )
If two numbers b and c have the property that their difference b-c is integrally
divisible by a number m { i.e (b-c)|m is an integer}, then b and c are said to be
“congruent modulo m”. The number m is called the modulus, and the statement “b is
congruent to c (modulo m)” is written mathematically as
b ≡ c (mod m)
If b-c is not integrally divisible by m, we say “b is not congruent to c(modulo m)”
which is written
b
c(modm)
m).
The quantity b is sometimes called the “base”, and the quantity c is called the
“residue or remainder”.
( Wikipedia)
Definition: If m  0, is a positive integer and a, b  Z; then we say a is congruent to
b modulo m if m|a-b.
Notation: In a ≡ b (mod m), the positive integer m is called a modulus
Examples:
45 ≡ 3 mod 6 ie. m|a-b
6
1
=
45  3 7
72 ≡ 0 mod 12 i.e m|a-b
12
1
=
72  0 6
-27 ≡ 0 mod 4
The idea of congruence and the notation a ≡ b(mod m) are due to Carl Friedrich
Gauss (1777-1855).
PROPERTIES OF CONGRUENCES MODULO M
Let a ≡ a’(mod m) and b=b’(mod m), then important properties of congruences
include the following;
1) Equivalence: a ≡ b(mod 0)  a ≡ b.
a
b(modm)
2) Determination: either a ≡ b(mod m) or
3) Reflexivity: a ≡ a(mod m)
4) Symmetry: a ≡ b(mod m)  b ≡ a(mod m)
5) Transivity: a ≡ b (mod m) and b ≡ c(mod m)  a ≡ c (mod m)
6) a+b ≡ a’+b’(mod m)
7) a-b ≡ a’-b’(mod m)
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8) ab ≡ a’b’(mod m)
9) a ≡ b(mod m)  ka ≡ kb(mod m)
10) a ≡ b(mod m)  an ≡ bn (mod m)
11) a ≡ b ( mod m1) and a ≡ b(mod m2)  a ≡ b (mod[m1,m2]), where [m1,m2] is the
least common multiple (LCM)
m
12) ak ≡ bk(mod m)  a ≡ b (mod
), where (k,m) is the greatest common
( k , m)
divisor (GCD).
13) If a ≡ b ( mod m), then p(a) ≡ p(b)(mod m), for p(x) a polynomia.
Theorem
If a,b,c and d  Z, then:
1) a ≡ b (mod m) iff b ≡ a (mod m) iff b – a ≡ 0 ( mod m)
2) If a ≡ b ( mod m ) and b ≡ c, then a ≡ c ( mod m)
3) If a ≡ b ( mod m) and d|m, d  0, then a ≡ b(mod d)
4) If a ≡ b(mod m) and c  0, then ac ≡ bc (mod mc)
5) If a ≡ b (mod m) and c ≡ d (mod m), then a+c≡ b+d (mod m)
6) If a ≡ b (mod m) and c ≡ d (mod m), ac ≡ bd(mod m)
Theorem (Cancellation Law)
Let m be any fixed modulus and suppose ab ≡ ac (mod m).
Then b ≡ c (mod m/d), where d = (a, m).
In particular, if a and m are relatively prime, then ab ≡ ac(mod m) implies
b ≡ c(mod m).
Propositions
1. Cancellation
If gcd(c, n)=1 and ac ≡ bc (mod n), then a ≡ b (mod n)
2. Units
If gcd(a,n)=1, then the equation ax ≡ b(mod n) has a solution, and the solution is
unique modulo n.
3. Solvability
The equation ax ≡ b(mod n )has a solution iff gcd(a,n) divides b.
Algorithm (Inverse Modulo n)
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Suppose a and n are integers and gcd(a,n)=1. The algorithm finds an x such that ax
≡ 1 (mod n)
Procedure: Compute extended GCD using Extended Euclidean Algorithm to
compute integers x,y such that ax + ny = 1
Example: Find an integer 37x ≡ 1(mod 101)
Solution
37x ≡ 1 (mod 101)
Step 1: Forming the equation
37x + 101y = 1
Step 2: Finding gcd = 1
Using the Extended Euclidean Algorithm,
101= 2 × 37 + 27 …………………
37 = 1 × 27 + 10………………….
27 = 2 × 10 +7 .……………………
10 =1 × 7 + 3 ……………………..
7 = 2 × 3 + 1……………………..
Thus gcd(101,37) = 1
(i)
(ii)
(iii)
(iv)
(v)
Step 3: Working through the following steps (i) ,(ii),(iii),(iv) and finally (v),
backwards;
i. 27= 101 – 2(37)
ii. 10= 37 – 1(27)
= 37 – 1[101  2(37)]
= 37 – 1(101) + 2(37)
= 101 + 3(37)
i.e Substituting the value of 27 in (i) above.
iii.
7 = 27 – 2(10)
= 101 – 2(37)  2[101 + 3(37)]
i.e. Substituting the final values of 27 and 10 in (i) & (ii) above
= 101  2(37) + 2(101) – 6(37)
= 3(101)  8(37)
iv.
3 = 10  1(7)
= 101 + 3(37) – 1[3(101)  8(37)]
i.e. Substituting the values of 10 and 7 in (ii) and (iii) above
= 101  3(101) + 3(37) + 8(37)
= 4(101) + 11(37)
v.
1=7–2×3
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= 3(101) - 8(37)  2[4(101) + 11(37)]
= 3(101) +8(101) – 8(37) -22(37)
= 11(101) -30(37)
Hence 37x + 101y ≡ 30(37) + 11(101)
 x = 30 is a solution to 37x ≡ 1(mod 101).
FORMATIVE EVALUATION 13:
Group Work: Linear Equations modulo n
1. Check on the proof of the cancellation law, Units and
Solvability in Stein 2005, pages 21- 26
2. How to solve ax ≡ 1 (mod n). Stein 2005, pages 29 –
31. Hence solve question 2.9 in exercise 2.6 on page
39
RESIDUE CLASSES
The number b in the congruence a ≡ b (mod m) is called the residue of a(modm).
Common Residue
The value of b, where a ≡ b(mod m) taken to be nonnegative and smaller than m.
Minimal Residue
The minimal residue of a(mod m) is the value of b or b-m, whichever is smaller in
absolute value, where a ≡ b(mod m). If m=2b 9so that b=|b-m|, then the minimal
residue is taken as – b. the table below illustrates the common (positive) and
minimal residues for 0, 1, 2, and 3(mod 4)
n
Common Residue n(mod 4)
Minimal Residue n(mod 4)
0
0
0
1
1
1
2
2
-2
3
3
-1
Example
To find 3713 (mod 17).
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Solution:
37 ≡ 3
37² ≡ 3² ≡ 9 ≡ -8
374 ≡ 81≡ ≡ - 4
378 ≡ 16 ≡ -1,
So 3713 ≡ 371+4+8 ≡ 3( -4 )( -1) ≡ 12(mod 17).
REDUCED RESIDUE SYSTEM
Any system of (n) integers, where (n) is the totient function, representing all the
residue classes relatively prime to n is called a reduced residue system.
RESIDUE CLASS
The residue classes of a function f(x) mod n are all possible values of the residue
f(x) (mod n).
Example:
Residue classes of x² (mod 6) are { 0,1,3,4} since
0² ≡ 0 (mod 6)
1² ≡ 1 (mod 6)
2² ≡ 4 (mod 6)
3² ≡ 3 (mod 6)
4² ≡ 3 (mod 6)
5² ≡ 1 (mod 6), are all the possible residues.
A complete residue system is a set of integers containing one element from each
class, so {0,1,9,16} would be a complete residue system for x² (mod 6).
( Wolfram MathWorld )
Definitions:
1) If a ≡ b (mod m), then b is called a Residue of a mod m.
2) A set { x1, x2, x3,……….. xm } is called a Complete residue system(mod m) if  n,  x
 n ≡ xi(mod m)
3) The congruence class (residue class) of n(mod m) is the set {n+mx|x Z }.
4) A reduced residue system (mod m) is a set of integers ri with (ri ,m)=1  for any n
with (n,m)=1  ri  n ≡ ri (mod m)
EULER’S  - FUNCTION
Definition: Arithmetic Function ( f )
An arithmetic function is a function whose domain is the set of positive integers e.g if
a function f(p)= p for p = 1,2,3,4,…..assigns only positive values of the root, then we
say that the function is an arithmetic function.
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Definition: Multiplicative
A function G is multiplicative if G( pq) = G(p) G(q) whenever p and q are relatively
prime positive integers, and completely multiplicative if G(pq)=G(p)G(q) for all
positive integers p and q.
Definition: Euler’s  - Function ( Euler’s totient function)
The symbol  - (phi) is used to represent the Euler function.
 p>1, let  (p) designate the number of positive integers less than p and relatively
prime to p.
Example:  (15)= 8 ie there are 8 positive integers, 1,2,4,7, 8,11, 13, 14 less than
and relatively prime to 15. If given  (1) = 1, then  is an arithmetic function. This
function is called the Euler’s  function or Euler’s Totient function (Leonhard Euler
1701 – 1783, Swiss mathematician)
Properties of 
1
)
p
 is multiplicative ie  (pq) = (p).(  (q).
1. For any prime p,  (p) = p -1 =p(1-
Theorem
 (m) of the m distinct residue classes mod m are relatively prime to m, which is the
number of integers 0  r  m.
Fermat’s Little Theorem
If p is a prime and a is any integer, then
1. ap ≡ a(mod p).
2. If a and p are relatively prime, then ap-1 ≡ 1(mod p).
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Theorem
The system of congruences
x ≡ a(mod m)
x ≡ b(mod n)
Is solvable if and only if (m, n) divides b – a. In the case where a solution xo exists, a
number x is also a solution and if only x ≡ xo (mod[m,n]), where [ m, n] is the least
common multiple of m and n.
(Kirch, 1974 )
Reading:
1. Elementary Number Theorem, By Stein, October 2005,
pages 21- 37
2. Attempt exercise on page 38 No. 2.1,2.2,2.4(b)
3. Congruences MIT Units 5 and 6, pages 1 & 2 in each
4. Elementary Number Theory, By W. Edwin Clark, page 58
- 80
FORMATIVE EVALUATION 15:
Group Work:
m
)
( a, m)
2. Pove the proposition: If b ≡ c (mod m), then (b,m) ≡ (c,m).
1. Prove the theorem: ax ≡ ay(mod m) iff x ≡ y (mod
Refer: MIT notes, Congruences 21 Feb 2003, page 1& 2.
PRIMITIVE ROOTS
A primitive root of a prime p is an integer g such that g (mod p) has modulo order p1.
Generally, if gcd(g,n)=1 { g and n are relatively prime) and g is of modulo order (n)
modulo n where (n) is the totient function, then g is a primitive root of n.
If n has a primitive root, then it has exactly [ (n)] of them, which means that if p is
prime number, then there are exactly (p-1) incongruent primitive roots of p.for
n=1,2,3,…., the first few values of [ (n)] are 1,1,1,1,2,1,2,2,2,2,4,2,4,2,4,4,8.
N has a primitive root if it is of the form 2,4,pa or 2pa , where p is an odd prime and a
1. the first few n for which primitive roots exist are
2,3,4,5,6,7,8,9,10,11,13,14,17,18,19,22,…….., so the number of primitive roots of
order n for n=1,2,….are 0,1,1,1,2,1,2,0,2,2,4,0,4,……. The smallest primitive root for
the first few primes p are 1,2,2,3,2,2,3,2,5,2,3,2,6,3,5,2,2,2……..
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Table of the primitive roots for the first few n for which a primitive root exists.
n
g(n)
2
1
3
2
4
2
5
2,3
6
5
7
3,5
9
2,5
10
3,7
11
2,6,7,8
13
2,6,7,11
The largest primitive roots for n=1,2,…… are 0,1,2,3,5,5,0,5,7,8,0,11,………
(Wolfram Mathworld)
Let m be a positive integer. Let a be any positive integer relatively prime to m and let
k be the smallest positive integer such that a k ≡ 1 (mod m).The number k is called
the exponent to which a belongs modulo m.
Example:
7 belongs to the exponent 2 modulo 4 since 7² ≡ 1 (mod 4)
but71
1(mod 4)
Theorems:
1) If k is the exponent to which a belongs modulo m, then k divides (m).
2) For any prime p there are exactly (p – 1)incongruent primitive roots modulo p.
3) If p is any prime and g is any primitive root modulo p, then the powers g,
g2,….,gp-1 form a reduced system of residues modulo p.
4) Let m be any integer greater than 1. Primitive roots exist modulo m if and only if
m=2, m=4, m=pn, m=2pn where p is an odd prime.
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FORMATIVE EVALUATION 16:
Group Work: Study the proof of the theorem of primitive
roots. Make sure you can proof this theorem during
examinations
Refer: Elementary Number Theorem, By Stein, October
2005, page 36.
Pythagorean Triples
History of Pythagorean Triples
The study of Pythagorean triples began long before the time of Pythagoras.
Babylonians and ancient Egyptians used the triples
Pythagorean Triples
Figure 1: Pythagorean Triangle
c
a2 + b 2 = c 2
b
a
Examples of Pythagorean Triples
 32 + 4 2 = 5 2
 52 + 122 = 132
 82 + 152 = 172
 282 + 452 = 532
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Table 1:
S/No
a” Odd”
b “ Even”
c “Odd”
Equation
1
3
4
5
32 + 42 = 5 2
2
5
12
13
52 + 122 = 132
3
7
24
25
72 + 242 = 252
4
9
40
41
92 + 402 = 412
5
11
60
61
112 + 602 = 612
6
15
8
17
152 + 82 = 172
7
21
20
29
212 + 202 = 292
8
33
56
65
332 + 562 = 652
9
45
28
53
452 + 282 = 532
PRIMITIVE PYTHAGOREAN TRIPLES
Definition:
A Primitive Pythagorean triple is a triple of numbers (a,b,c) so that a, b and c have
no common factors and satisfy a2 + b2 = c2
Observations on the Pythagorean Triples ( Table 1)
 One of a and b is odd and the other even and it seems c is always odd.
 Taking a to be odd and b to be even, then for a2 + b2 = c2
We can find a as a2 = c2  b2 = (c – b) ( c + b).
Examples
3,4,5
 32 = (52 – 42 )=(5 –4)(5 + 4) = 1 × 9 = 12 × 32
5,12,13  52 = (132 – 122)=(13 –12)(13 + 12) = 1 × 25 = 12 × 52
7,24,25  72 = (252 – 242)=(25 –24)(25 + 24) = 1 × 49 = 12 × 72
15,8,17  152 = (172 – 82)=(17 –8)(17 + 8) = 9 × 25 = 32 × 52
 From observations, it seems
1. (c – b) and (c + b) are always Prime odd squares
2. (c – b) and (c + b) have no common factors.
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Pythagorean Triples and the Unit Circle
a 2
b
) + ( )2 = 1
c
c
a
b
It follows the rational numbers   and (   ) is a solution to the equation of a
c
c
circle
x2 + y2 = 1, which describes a circle of radius 1 with center (0, 0) on the Cartesian
plane.
Given a2 + b2 = c2, dividing through by c2  (
List of Pythagorean Triples
( a, b, c)
( a, b, c)
( a, b, c)
( a, b, c)
3,4,5
64,1023,1025
84,13,85
96,2303,2305
5,12,13,
68,285,293
84,187,205
100,621,629
7,24,25
63,1155,1157
84,437,445
100,2499,2501
9,40,41
72,65,97
84,1763,1765
15,8,17
72,1295,1297
88,105,137
21,20,29
76,357,365
88,1935,1937
35,12,37
76,1443,1445
92,525,533
45,28,53
80,39,89
92,2115,2117
63,16,65
80,1599,1601
96,247,265
Fermat’s Last Theorem
The Diophantine equations x + y = z and x2 + y2 = z2 have infinitely many solutions.
In 1637, Fermat wrote that it is impossible to write a (positive) cube as a sum of two
cubes for example, x3 + y3 = z3 or a fourth power x4 + y4 = z4, as a sum of two fourth
powers, in fact any power above the second as a sum of two like powers i.e he
wrote the Diophantine equation xn + yn = zn has no positive solutions for n  3.
The proof for this assertion has been pending for 358 years until the year 1993,
when Andrew Wiles brought forth a tentative proof. Five years later the proof was
confirmed.
Internet Activity (All sources last accessed 03.11.06)
Explore the history of the proof of Fermat’s last theorem at the MacTutor archive at
the University of St. Andrews in Scotland, UK.
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The Theorem and its proof:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html
Pierre de Fermat:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Fermat.html
Andrew Wiles:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Wiles.html
Interview of Andrew Wiles with Nova Magazine:
http://www.pbs.org/wgbh/nova/proof/wiles.html
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UNIT 2 ( 60 Hours): THEORY OF CONGRUENCE AND QUADRATIC
FIELDS
Summary of Unit 2: ( Multiple Activities)
Unit two of the number theorem assumes unit one as a pre-requisite. It illustrates the
field of integers, squares and quadratic residues. It introduces the Legendre symbol,
Gauss lemma and quadratic reciprocity law. It analyses the quadratic fields, and
applies Euclid’s algorithm and unique factorization of Gaussian integers. It analyses
the arithmetic of quadratic fields and application of Diophantine equations, and
concludes with Pell’s equation and units in real quadratic fields. The unit is
organised with multiple activities for the learners and has formative evaluations after
every subtopic.
THE FIELD OF INTEGERS ( MOD P), SQUARES AND QUADRATIC RESIDUES
Definition
If xn = a(mod m), has a solution, where a and m are relatively prime, then a is called
an nth power residue modulo m.
If the congruence has no solution, then a is called an nth power non-residue modulo
m.
Reading:
1. Solving equations Modulo Primes, MIT- 14- Notes, pages 1 & 2
2. More on solving equations , modulo primes, MIT Unit 15, pp 1 & 2.
3. Quadratic Residue Symbol, MIT – 16- Notes, pages 1 & 2.
4. Elementary Number Theory, By W. Edwin Clark, page 76 - 80
QUADRATIC RECIPROCITY, THE QUADRATIC RECIPROCITY LAW AND THE
LEGENDRE SYMBOL
Definition:
The linear equation a ≡ b (mod n) has a solution if and only if gcd(a.n)divides b.
Quadratic reciprocity searches for a criterion for whether or not equation
ax² + bx + c ≡ 0 (mod n).
Definition
Fix a prime p. An integer a not divisible by p is quadratic residue modulo p if a is a
square modulo p; otherwise, a is a quadratic nonresidue.
Definition: Legendre Symbol
Let p be an odd prime and let a be an integer coprime to p. Set
+1 if a is a quadratic residue
1
otherwise
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a
  =
 p
a
We call this symbol   the Legendre symbol. This notation is well entrenched in
 p
the literature even though it is also the notation for “ a divided by p”.
a
Note:   only depends on a(modp), it makes sense to define
 p
a
  for a є Z / pZ to
 p
 a~ 
be   for any lift a~ of a to Z.
 p
Legendre Symbol of 2
Definition:
Let p be odd prime.
2
Then   =
 p
+1 if p ≡  1 (mod 8)
1
if p ≡  3 (mod 8)
FORMATIVE EVALUATION 17:
Group Work: Check for the proof
1. Legendre assertion from Number Theory by Stein page
67
2. Quadratic reciprocity from Number Theory by Stein
page 68 – 72
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EULER’S CRITERION, GAUSS LEMMA AND THE QUADRATIC RECIPROCITY
LAW
EULER’S CRITERION
Let p be an odd prime and a an integer not divisible by p. Euler used the existence
a
of primitive roots to show that   is congruent to a(p – 1)/2 modulo p.
 p
a
Eulers criterion: We have   ≡ 1 if and only if a(p – 1)/2 = 1 ( mod p)
 p
Let p be an odd prime and let a be an integer not equal to 0 ( mod p). From the
numbers
a, 2a, 3a,………,
p 1
 p p
a and reduce them modulo p to lie in the interval 
,  .
2
 2 2
a
Let  be the number of negative numbers in the resulting set. Then   =(1)  .
 p
FORMATIVE EVALUATION 18:
Group Work: Study the proof of
1. Eulers criterion, Number Theory by Stein page 62.
2. Gauss lemma, Number Theory by Stein page 64.
3. Quadratic reciprocity using Gauss sums, Number
Theory by Stein page 71.
Refer:
Reading
1. Solving Equations Modulo Primes MIT Unit14, pages 1& 2
2. More on quadratic Residues, MIT Units17,18,19,& 20, pages
1 & 2 each
3. Elementary Number Theory, By Stein, October 2005, pp 59
– 72
Complete questions 4.1, 4.2 on page 74
1. Elementary Number Theory, By W. Edwin Clark, page 24 25
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EVALUATING THE QUADRATIC CHARACTER BY THE RECIPROCITY LAW
Theorems:
1. Let p be an odd prime and let a be relatively prime to p. Then 2 is a quadratic
residue for all primes of the form 8n  1; 2 is a quadratic non-residue for all
primes of the form 8n  3.
2. If p is prime and (a,p) = 1, then the congruence ax² + bx + c ≡ 0 (mod p) has at
most two incongruent solutions modulo p.
Reading:
1. Solving Equations Modulo Primes, MIT Units 14,17,18,19 &
20 pp 1 & 2 each.
2. Elementary Number Theory, By Stein,Oct 2005, pages 59-72
3. Elementary Number Theory, By W. Edwin Clark, page 58 - 74
PELL’S EQUATION AND UNITS IN REAL QUADRATIC FIELDS
Pell’s Equation
The Diophantine equation x2 –qy2 = 1 (q ≠ 0) is called Pell’s equation. Pell’s equation
focuses on positive integral values of x and y that satisfy the equation, assuming that
q is a positive integer. Assume that q is a positive integer and thus 1 is not the
difference of squares because q is not squared.
Fundamental Solution of Pell’s Equation
Let q be a positive integer which is not a square. By a positive solution of Pell’s
equation, we mean a pair of positive integers x 0 and y0 such that x02 + y02 =1. If the
equation has any solution in nonzero integers x0 and y0,then it has a positive
solution, namely x0 , y0. The positive solution x1 ,y1 which minimizes the quantity
x1 + y1 q is called the Fundamental Solution of Pell’s Equation.
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The table below gives the Computer generated solutions for Pell’s equation
qy2 = 1 for non-square integers q satisfying 1  q ≤ 52.
q
x
y
q
2
3
2
3
2
5
x
y
q
18 17
4
1
19 170
9
4
6
5
7
8
x
y
q
x
32 17
3
46 24335
3588
39
33 23
4
47 48
7
20 9
2
34 35
6
48 7
1
2
21 55
12
35 6
1
50 99
14
8
3
22 197
42
37 73
12
51 50
7
3
1
23 24
5
38 37
6
52 649
90
10 19
6
24 5
1
39 25
4
11 10
3
26 51
10
40 19
3
12 7
2
27 26
5
41 2049
320
13 649 180
28 127
24
42 13
2
14 15
4
29 9801
1820
43 3482
531
15 4
1
30 11
2
44 199
30
17 3
8
31 1520
273
45 161
24
x2 –
y
Theorem
Let x1, y1 be the fundamental solution of Pell’s equation (for given q). Then x’, y’ is a
positive solution if and only if
X ’+ y ’q=(x1+y1q)n
for some positive integer n.
Example Of Solving Pell’s Equation
Solve the Diophantine equation x² - 5y²=1
Solution
Step 1: The Diophantine equation is called Pell’s equation, e.g. x² -qy²=1.
Step 2: From the above table of fundamental solutions of Pell’s equation, we note
that when q = 5, x = 9, and y = 4.These are the fundamental solutions of Pell’s
equation.
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Step 3: From the above theorem, Pell’s equation has infinitely many solutions.
Substituting q = 5, x = 9 and y = 4, we get:
(9 + 45)² = 161 + 725
And
(9 + 45)³ = (9 + 45)(161 + 725) = 2889 + 12925
The next two larger positive solutions are x = 161, y = 72 and x = 2889, y = 1292
FORMATIVE EVALUATION 19:
Exercise: Solutions Involving Pell’s Equations
Find all positive solutions of:
1. x² - 2y² = 1
2. x² - 3y² = 1
Solutions
1. (3,2), (17,12), (99,70)….are the first few solutions.
2. (2,1), (7,4), (26,151) …are the first few solutions.
Reading:
1. Solving Equations Modulo Primes, MIT Units 14,17,18,19
& 20, pp 1 & 2 each.
2. Elementary Number Theory, By Stein, October 2005, pp
59-72
*One relevant image must be inserted here.
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15. Synthesis of the Module
Module Writing Tip. Module Developers should write a synthesis of the most important content learners are expected to know
at the end of the module. This summary of the content could take the form of a written text (maximum length : 300 words).
*One relevant image must be inserted here.
At the end of this module learners are expected to know the properties of integers,
divisibility of integers, prime numbers and their distribution. The application of
divisibility in Euclidean algorithm forms the background that leads to solutions of
linear Diophantine equations. The learners should be able to solve linear
Diophantine equations using Euclidean algorithm.The pythagorean triples as relates
to the Fermat’s theorem is a fundamental area that learners should know. Unit one
of Number Theory is covered through various illustrated examples that learners can
follow without difficulty. It is recommended that learners attempt the formative
evaluations given to assess their progress in the understanding of the content.
Learners should take time to check the recommended reference material in CD’S,
attached open source materials and the recommended websites. Most importantly,
learners are encouraged to read the contenet widely and attempt the questions after
each topic. Unit two of the module takes learners through residues, their properties
and the quadratic reciprocity. Euler’s criteria and the notation of the legendre symbol
are important .The learners should be able to define the Norm and it’s application in
the unique factorisation of the Gaussian integers. Learners should be able to prove
lemmas in Gaussian integers. Unit two has various lerning activities to aid learning
and learners are advised to master the content of the various sub-topics and assess
themselves through the formative evalutaions. Failure to answer the formative
assessments should be a positive indicator that learners should revise the subtopics before prcgressing to other sub-topics. Throughout the number theory
module, the activities are formulated to enable easy learning. The tasks given under
the different learning activities demands that you demonstrate a high level of ICT
skills competency. The learning objectives are well stated in the beginning of the
module and should guide learners in the level of expectations of the module. The
final part of the module deals with Pell’s equation that requires learners to project the
skills mastered in Diophantine equations to understand this special form of the
Diophantine equation. The summative evaluation will be used to judge the learners
mastery of the module. It is recommended that learners revise the module before
sitting for the final summative evaluation.
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6. Summative Evaluation
Module Writing Tip. A summative evaluation can take different forms : a test (multiple choice, short answers, etc.), a project, a
written production, a problem solving task, etc. The summative evaluation is usually what is used to provide students with a
final mark for the module. This section is therefore designed to provide information to determine the amount of learning by a
student at the end of the module. Module Developers should keep in mind that the summative evaluation must be conducted in
a distance education context. They should also carefully consider what should be evaluated, and how it should be evaluated.
Instructions provided to learners in a final evaluation must be clear, concise and well written. An answer key must be provided
by Module Developers. For multiple choice exams, a significant database of questions (3 to 5 for each topic) must be provided
so that exams can vary significantly from student to student. Also, Module Developers must indicate how learners will submit
their answers to the instructor. Will the summary evaluation be emailed to the instructor ? Will it be submitted online ? Will there
be an online test ? Will there be an on-site evaluation ? Finally, summative evaluations should include opportunities for
students to respond to the module to provide feedback to the instructor.
SUMMATIVE ASSESSMENT ( Answer any THREE questions- 60%)
1. Use Euclidean algorithm to compute gcd of
(i)
m= 25 174, n=42 722
(ii)
m=7472, n=2464
(iii)
m=455, n=1235
2. Prove by induction that:
(i). Prove that 13 + 23 + 33 + 43 + ..…+ n3=
(ii). Prove that a+ar+ar2+…+arn=
n 2 (n  1) 2
4
a(1  r n 1 )
1 r
, n>0
3. a). Prove the Cancellation proposition: If gcd(c,n)=1 and ac ≡ bc ( mod n),
then a ≡ b (mod n)
b) Solve 17x ≡ 1(mod 61)
4. The table below shows some fundamental solutions of Pell’s Equation.
Value of q
Value of x
Value of y
6
5
2
10
19
6
14
15
4
Use the table to solve the following Pell’s equations:
x2 – 6y2=1
x2 – 14y2=1
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5. Solve for x and y in the Diophantine equation
2261x+1275y= gcd(2261,1275).
17. References
Module Writing Tip. Module Developers should include all relevant references (minimum 10) for the content of the module, as
well as all references used in writing the module. All references should be written using the APA style guidelines (see Annex 5
or 6, or consult the American Psychological Association Web site : http://www.apa.org/books/4200061.html). References that
do not comply with the APA style guidelines will not be accepted.
References
1) Elementary Number Theory, By W. Edwin Clark, University of South Florida,
2003
2) http://www-history.mcs.st-andrews.ac.uk/Biographies/Wiles.html
3) Notes on Algebraic Numbers, By Robin Chapman, 2002
4) Algebra and Number theory, By A. Baker, University of Glasgow,2003
5) http://www.pbs.org/wgbh/nova/proof/wiles.html
6) Prime factorization, By William Stein, Havard University,2001
7) Lecture notes, By William Stein, Havard University, 2001
8) http://www-history.mcs.st-andrews.ac.uk/Biographies/Wiles.html
9) Elementary Number Theory, By Allan M. Kirch, Intext Educational Publishers,
New York, 1974.
10) http://www-history.mcs.st-andrews.ac.uk/Biographies/Wiles.html
11) Elements of Abstract & Linear Algebra, By E.H Connell, Coral Gables, Florida,
USA.
12) MIT Open Courseware, Theory of Numbers, Spring 2003, Prof. Martin Olsson
13) http://www-groups.dcs.st-and.ac.uk~history/HistTopics/Fermat’s last
theorem.html
14) http://www.bbc.co.uk/education/asguru/maths/13pure/01proof/01proof/05inducti
on/index.shtml
15) http://en.wikipedia.org/wiki/Number_Theory
16) http://mathworld.wolfram.com/NumberTheory.html
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18. Student Records
Module Developer Writing Tip. Though for most modules the final mark (for one module) will be closely linked to the
summative evaluation, it is often wise to mark or give points for the completion of other activities or formative evaluations.
Module Developers are therefore required to provide a clearly laid out “My Records” spreadsheet page that includes:
- Organized columns for entry of “future students” ;
- Organized columns for entry of all required marks ;
- Calculated columns to indicate overall achievement.
- Module Developers should provide the name of the EXCEL file.
Name of the EXCEL file :
Mathematics Number Theory Student Records
19. Main Author of the Module
Module Developer Writing Tip. Module Developers should provide a brief biography (50-75 words), a picture, title and contact
information (email).
Name: Mr. Paul Chege ( B.Ed(Sc), M.Ed )
Contact: [email protected]
The module author is a teacher trainer at Amoud
University, Borama, Republic of Somaliland.
He has been a teacher trainer in Kenya, Republic of
Seychelles, and Somalia. He has been involved in
strengthening Mathematics and Sciences at secondary
and tertiary levels under the Japan International
Coorporation Agency (JICA) programme in fifteen African
countries.
He is married with three children.
20. File Structure
Module Developer Writing Tip. The file naming and structure must follow the AVU/PI Consortium template as defined and
explained by the AVU. Module Developers still need to provide the name of all the files (module and other files accompanying
the module).
Daily, each module will be loaded in the personal eportfolio created for each consultant. For this, training will be provided by
professor Thierry Karsenti and his team (Salomon Tchaméni Ngamo and Toby Harper).
Name of the module (WORD) file : Mathematics
Number Theory ( Word)
Name of all other files (WORD, PDF, PPT, etc.) for the module.
1.
2.
3.
4.
5.
6.
Number Theory Students Record ( Excel)
Marking Scheme for Summative Evaluation ( Word)
Number Theory Lecture Notes by Stein ( PDF)
Elementary Number Theory Textbook by Clarke (PDF)
Number Theory Textbook by Stein (PDF)
MIT-Theory of Numbers Lecture Notes and Exams ( PDF)
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Marking Scheme for the Summative Assessment
1 (i) 42 722 = 1 x 25 174 + 17 548
25 174 = 1 x 17 548 + 7 626
17 548 = 2 x 7 626 + 2 296
7 626 = 3 x 2 296 + 738
2 296 = 3 x 738 + 82
738 = 9 x 82
Hence the gcd ( 25 174, 42 722) = 82
(ii) 7472 = 3 x 2464 + 80
2464 =30 x 80 +64
80 = 1 x 64 + 16
64 = 4 x 16
Hence the gcd ( 7 472, 2 464) = 16
(iii) 1235 = 2 x 455 + 325
455 = 1 x 325 +130
325 = 2 x 130 + 65
65 = 2 x 65
Hence the gcd ( 455, 1235 ) = 65
2. Prove that 13 + 23 + 33 + 43 + ..…+ n3=
n 2 (n  1) 2
4
Proof:
Step 1: Technique of Induction
Mathematical induction proves by checking if a proposition hold’s for n=1, and
n= k+1 whenever it holds for n=k, then the proposition holds for all positive
integers n= 1,2, 3,……..
Step 2: Substitute n=1, in the equation:
13 =
12 (1  1) 2
 1=1 i.e It holds for n =1.
4
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Step 3: Assume that the formula holds for n= k
k 2 (k  1) 2
1 + 2 + 3 + 4 + ..…+ k =
4
3
3
3
3
3
Step 4: Proof that the formula holds for n= k+1.
13 + 23 + 33 + 43 + ..…+ k3 + (k+1)³ =
(k  1) 2 (k  1  1) 2
4
We write
13 + 23 + 33 + 43 + ..…+ k3=


k 2 (k  1) 2
4
k 2 (k  1) 2
(k  1) 2 (k  1  1) 2
+ ( k + 1)³ =
4
4
k 2 (k  1) 2  4(k  1) 3
(k  1) 2 (k  2) 2
=
Cancelling the 4 on both sides
4
4
 k 2 (k  1) 2  4(k  1) 3 = (k  1) 2 (k  2) 2
 (k  1) 2 [(k 2  4(k  1)] = (k  1) 2 (k  2) 2 Dividing by (k+1)²
 k² +4k +4 = k² +4k +4 ( Proved)
This is proof by induction
(ii). Prove that a+ar+ar2+…+arn=
Module Development Template
a(1  r n 1 )
, n>0
1 r
65
Proof:
Step 1: Technique of Induction
Mathematical induction proves by checking if a proposition hold’s for n=1, and
n= k+1 whenever it holds for n=k, then the proposition holds for all positive
integers n= 1,2, 3,……..
Step 2: Substitute n=1, in the equation:
a+ar=
a(1  r )(1  r )
a (1  r 11 )
a (1  r 2 )
=
=
= a + ar i.e It holds for n =1.
1 r
1 r
1 r
Step 3: Assume that the formula holds for n= k
a+ar+ar2+…+ark=
a(1  r k 1 )
1 r
Step 4: Proof that the formula holds for n= k+1.
a(1  r k 11 )
a+ar+ar2+…, + ark +ark+1=
1 r
We write
a+ar+ar2+…+ark=
a(1  r k 1 )
1 r

a(1  r k 1 )
a(1  r k  2 )
k+1
+ ar =
1 r
1 r

a(1  r k 1 ) ar k 1 (1  r )
a(1  r k  2 )
+
=
1 r
1 r
1 r

a(1  r k 1 )  ar k 1 (1  r )
a(1  r k  2 )
=
1 r
1 r
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66

a  ar k  2 )
a(1  r k  2 )
=
Opening brackets and simplifying
1 r
1 r

a(1  r k  2 )
a(1  r k  2 )
=
( Proved)
1 r
1 r
This is proof by induction
3. a) By definition, n|ac –bc = (a-b)c and since gcd(n,1), it follows that n|a-b, so a ≡
b(mod n).
b).Use Euclid’s algorithm to solve 17x ≡ 1(mod 61)
Solution:
Step 1: Find x and y such that 17x + 61y = 1
Using Euclid’s algorithm
61= 3 x 17 + 10
17 = 1 x 10 + 7
10 = 1 x 7 + 3
7 =2x3+1
Step 2:
10 = 61 – 3 x 17
7 = 17 – 1 x 10 = 17 -1[61 -3(17)]
= 17 – 1(61) +3(17)
=4(17) – 61
= -61 + 4(17)
3 = 10 – 1(7)
=61 – 3 x 17 – 1(7)
= 61 – 3 x 17 – 1[-61 + 4(17)]
= 61 + 1(61) -3(17) - 4(17)
=2(61) – 7(17)
1 =7 – 2x3
= -61 + 4(17) – 2[2(61) – 7(17)]
= -61 - 4(61) +4(17) + 14(17)
= -5(61) + 18( 17)
Hence 17x + 61y ≡ 18( 17) - 5(61)
 x = 18 is a solution to 17x ≡ 1(mod 61)
4. a)From the table, when q=6, then x=5 and y= 2
 ( 5 + 2 6)² = 25 + 206 +24 = 49 +206
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 x = 49 and y = 20 are the fundamental solutions of Pell’s equation
Larger solutions are:
(5 + 26)³ = (5 + 26 )( 49 +206) = 485 + 1986.
Hence x=485 and y = 198.
b) From the table, when q=14 then x=15 and y = 4.
 ( 15 + 4 14)² = 225 + 12014 +224 = 449 +12014
 x = 449 and y = 120 are the fundamental solutions of Pell’s equation
Larger solutions are:
(15 + 414)³ = (15 + 414 )( 449 +12014) = 13 455 + 359614.
Hence x=13 455 and y = 3596.
5. Solve for x and y in the Diophantine equation
2261x+1275y= gcd(2261,1275).
Step1:gcd(2261,1275)
2261=1 x 1275 + 986
1275 = 1 x 986 + 289
986 = 3 x 289 + 119
289 = 2 x 119 + 51
119 = 2 x 51 + 17
Hence gcd ( 2261,1275) = 17
Substituting the gcd in equation
2261x + 1275y = 17
Applying back substitution:
17 = 119 – (2 x 51)
=119 – 2( 289 -2(119))
=119 -2(289) +4(119)
=5(119) – 2(289)
=5(119)-2(1275-1(986))
=5(986 -3 x 289) -2(1275 – 1(986))
=5(986) – 15(289) –2(1275) +2(986)
=7(986) – 15(1275 – 1(986)) – 2(1275)
=7(986) – 15(1275) +15(986) – 2(1275)
=22(986) -17(1275)
=22(2261 – 1(1275)) -17(1275)
=22(2261) – 22(1275) -17( 1275)
=22(2261) – 39(1275)  2261x + 1275y
Hence x = 22 and y = -39.
Module Development Template
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