MA 4203 Number Theory
Prerequisite: Void.
Well Ordering Principle and Induction, Divisibility, properties of integers and
prime numbers, fundamental theorem of arithmetic
Congruences, solutions of Congruences, Congruences of degree one, Congruences
of hig

2.2. General properties of congruences
In this module we now discuss some general properties of congruences.
Common nonzero factors cannot always be cancelled from both members of the
congruence as they can be done in equations.
For example, both members

2.4. Simultaneous Linear Congruences and the
Chinese Remainder Theorem
Having considered a single linear congruence, it is natural to discuss the problem
of solving a system
(
)
(
)
)
.
(
of simultaneous linear congruences.
It may be noted that a sy

1.8. The Euclidean Algorithm
The gcd of two integers can be found by listing all their positive divisors and
taking out the largest common divisor. Clearly, it is a cumbersome for large
numbers. A more efficient process, involving repeated application of

2.4. Simultaneous Linear Congruences and the
Chinese Remainder Theorem
Exercise:
1. Solve the following system of linear congruences:
3 , 2
5 , 3
7
Ans:
a) 1
b) 5
11 , 14
29 , 15
31
Ans:
c) 5
6 , 4
11 , 3
17 Ans:
d) 2 1
5 , 3

2.2. General properties of congruences
Exercise
1. Prove that
2. Find the remainders when
is divisible by .
and
are divided by . (Ans: and )
3. What is the remainder when the sum
is divided by . (Ans: )
4. Using the congruence theory prove the following:

1.9. The Diophantine Equation
An equation in one or more unknowns with integer coefficients admitting
integer solutions is called a Diophantine equation and the study of such
equations is known as Diophantine analysis.
The equation
for Pythagorean triples

2.3. Linear Congruences
We consider polynomials
with integer coefficients, so that these polynomials
will be integers when is an integer.
Let
be a given positive integer. Let
=
+
+
+ +
0
be a polynomial with integer coefficients of degree , i.e.,

1.6. Fundamental Theorem of Arithmetic
Theorem 1: Fundamental Theorem of Arithmetic (or unique Factorization
Theorem)
Every integer
can be represented as a product of prime factors in only one
way, apart from the order of the factors.
Proof: We prove the

UNIT-2
2.1. The Theory of Congruences
Another approach to divisibility related questions is through the arithmetic
of remainders or the theory of congruences, as it is now commonly known.
The concept and notation was first introduced by the German
mathema

1.5
Prime and Composite Numbers
We start with prime numbers. Prime numbers are crucial for understanding of
natural numbers. Infact they are the building blocks of natural numbers.
Prime number:
An integer is called a prime number if
are and . That is, a

1.7. The Division Algorithm
Theorem 4 of Module 1.6 gives a practical method of computing the gcd of two
integers and , when the prime power factorization of and are known.
Considerable computations are required to obtain these prime-power
factorizations.

1.9. The Diophantine Equation
Exercise
1. Which of the following Diophantine equations cannot be solved?
(i)
(ii)
(iii)
(Ans: (i) No
(ii) Yes
(iii) No)
2. Determine all solutions in the integers of the following Diophantine equations:
(i)
(ii)
(iii)
3. De

1.4
Greatest common divisor (cont) and
Least Common multiple (l.c.m)
The g.c.d of more than two integers: The g.c.d of
defined inductively by the relation
,
,
=
,
,
integers
,
,
is
,
This number is independent of the order in which the
appear.
If =
, ,
,

1.1.
Well-Ordering Principle and Induction
Introduction
The theory of numbers is primarily concerned with the properties of the natural
numbers
1,2,3,4, ,
also called the counting numbers or the positive integers.
The theory is not strictly confined to j

1.2&1.3
Divisibility in Integers and Greatest Common Divisor.
An integer divides an integer (or is a divisor of or is a factor of ),
written as | , if =
for some integer . We also say that is a multiple of
if is a divisor of .If does not divide , then we