Math 135: Lecture 12: Modular Arithmetic
Definition 12.1.
The
congruence class modulo m
of the integer
a
is the set of integers
[
a
]
m
=
{
x
∈
Z

x
≡
a
(mod
m
)
}
Example 12.2.
When
m
= 4
[0]
4
=
{
x
∈
Z

x
≡
0
(mod 4)
}
=
{
. . . ,

8
,

4
,
0
,
4
,
8
, . . .
}
=
{
4
k

k
∈
Z
}
[1]
4
=
{
x
∈
Z

x
≡
1
(mod 4)
}
=
{
. . . ,

7
,

3
,
1
,
5
,
9
, . . .
}
=
{
4
k
+ 1

k
∈
Z
}
[2]
4
=
{
x
∈
Z

x
≡
2
(mod 4)
}
=
{
. . . ,

6
,

2
,
2
,
6
,
10
, . . .
}
=
{
4
k
+ 2

k
∈
Z
}
[3]
4
=
{
x
∈
Z

x
≡
3
(mod 4)
}
=
{
. . . ,

5
,

1
,
3
,
8
,
11
, . . .
}
=
{
4
k
+ 3

k
∈
Z
}
Note that congruence classes have more than one representation.
In the example above,
and, in fact [0]
4
has
representations.
Defining
Z
m
We define
Z
m
to be the set of
m
congruence classes
Z
m
=
{
[0]
,
[1]
,
[2]
, . . . ,
[
m

1]
}
and we define two operations on
Z
m
,
addition
and
multiplication
, as follows:
[
a
] + [
b
] = [
a
+
b
]
[
a
]
·
[
b
] = [
a
·
b
]
Example 12.3
(Addition in
Z
4
)
.
+
[0]
[1]
[2]
[3]
[0]
[0]
[1]
[2]
[3]
[1]
[1]
[2]
[3]
[0]
[2]
[2]
[3]
[0]
[1]
[3]
[3]
[0]
[1]
[2]
Example 12.4
(Multiplication in
Z
4
)
.
·
[0]
[1]
[2]
[3]
[0]
[0]
[0]
[0]
[0]
[1]
[0]
[1]
[2]
[3]
[2]
[0]
[2]
[0]
[2]
[3]
[0]
[3]
[2]
[0]
Exercise 12.5.
Complete the addition table in
Z
5
+
[0]
[1]
[2]
[3]
[4]
[0]
[1]
[2]
[3]
[4]
Exercise 12.6.
Complete the multiplication table in
Z
5
+
[0]
[1]
[2]
[3]
[4]
[0]
[1]
[2]
[3]
[4]
1
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Math 135: Lecture 12: Modular Arithmetic
Zero in
Z
m
From the definition of addition and multiplication in
Z
m
∀
[
a
]
∈
Z
m
,
∀
[
a
]
∈
Z
m
,
One in
Z
m
From our definition of multiplication in
Z
m
∀
[
a
]
∈
Z
m
,
Identity
Given a set and an operation, an identity is, informally, “something that does nothing”. More
formally, given a set
S
and an operation designated by
◦
, an
identity
is an element
e
∈
S
so
that
∀
a
∈
S, a
◦
e
=
a
Example 12.7.
•
Set: integers; Operation: addition; Identity:
•
Set:
Q
 {
0
}
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 Fall '08
 ANDREWCHILDS
 Math, Addition, Congruence, Integers, Ring, zp, Zm

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