Download Basic Definitions and Notations (1)

Number Theory
Lecture 1
Divisibility and Modular Arithmetic
(Congruences)
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Peter Burkhardt
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Basic Definitions and Notations (1)
 N = {1,2,3,…}
denotes the set of natural
numbers
 Z = {…,-3,-2,-1,0,1,2,3,…}
denotes the set of integers
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Basic Definitions and Notations (2)
Divisibility (1)
Let a, b e
Z, a not equal to zero.
We say a divides b if there exists
an integer k such that
b

k

a
b  k a
b  k a
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Basic Definitions and Notations (3)
Divisibility (2)
In this case we write
a|b
Sometimes we say that:
 b is divisible by a, or
 a is a factor of b, or
 b is a multiple of a
b  k a
b  k a
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Basic Definitions and Notations (4)
Prime and Composite Numbers
A natural number p > 1 is called a prime
number, or, simply, prime, if it is divisible
only by itself and by 1.
P = {2,3,5,7,…}
denotes the set of prime numbers.
Otherwise the number is called composite.
b  k a
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Properties of Divisibility
 a|b  a|bc for each integer c
 a|b and b|c  a|c
 a|b and a|c  a|(bx + cy) for any x, y e
Z
 a|b and a, b not equal to zero  |a|  |b|
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Division with Remainder
Let m ,a e Z, m > 1.
Then, there exist uniquely determined
numbers q and r such that
a = qm + r
with
0r<m
Obviously, m|a if and only if r = 0.
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Congruences
Let a, b e Z, m e N. We say
a is congruent to b modulo m
if
m|(a-b)
and we write
a  b(mod m) or, shorter, a  b(m)
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Congruence and Division with
Remainder
a  b(mod m)
a  b(mod m)


Dividing a and b by m yields the same remainder.
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Basic Properties of Congruences
a, b, c  Z , m  N :
Reflexivit y : a  a(mod m)
Symmetry : a  b(mod m)  b  a(mod m)
Transitivi ty : a  b(mod m), b  c(mod m)  a  c(mod m)
That is, congruence is an equivalence relation.
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Modular Arithmetic
a, b, c, d  Z , m  N :
If a  b(mod m), c  d (mod m), then
Addition Rule : a  c  b  d (mod m)
Product Rule : a  c  b  d (mod m)
a, b, c, d  Z , m  N :
If a  b(mod m), c  d (mod m), then
Addition Rule : a  c  b  d (mod m)
Product Rule : a  c  b  d (mod m)
Power Rule : a k  b k (mod m) k  N
Power Rule : a  b (mod m) k  N
k
k
Factor Rule : a  c  b  c(mod m  c) for c  0
 Demonstration
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Little Fermat’s Theorem
Let a e Z, and p prime. If p does not
divide a, then
a
p 1
a p)1(mod p)
 1(mod
p 1
ae p
Z)we have
a For
 aall(mod
p
a  a(mod p)
p
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Have you understood?
How can you write the following statements
using congruences? (a, b, r e Z, m e N)
1. m|a
2. r is the remainder of a divided by m
Using congruences, give a sufficient condition
for
m|a if and only if m|b
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Practice (Handouts)
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