Chapter 4
Congruences
For this section, it is helpful if you keep “clock arithmetic” in mind. That is
14:00 hours is 2 o’clock. This will be considered doing arithmetic modulo 12.
We will just consider an
n
hour clock.
Defnition 5.
Let
n
∈
N
. Two integers
a
and
b
are said to be
congruent modulo
n
,wr
itten
a
≡
b
(mod
n
)
,
provided
n

(
a
−
b
).
The canonical example of a congruence comes from the division algorithm.
Example 5.
Let
a
∈
Z
and
n
∈
N
. Given the division algorithm we know that
there exists unique
q,r
∈
Z
with 0
≤
r<n
, such that
a
=
qn
+
r.
The de±nition of congruence gives
a
≡
r
(mod
n
)
.
Defnition 6.
If
a, b
∈
Z
and
a
≡
b
(mod
n
) we say that
a
is a
residue
of
b
modulo
n
.
Remark 4.
The above examples tells us that modulo
n
any integer
a
is con
gruent to one of
0
,
1
,
2
,...,n
−
1
.
These integers are called the
least nonnegative residues modulo
n
.
Defnition 7.
The set of integers
{
a
0
,a
1
,...,a
n
−
1
}
is called a
complete residue
system
modulo
n
provided
(i)
a
i
°≡
a
j
(mod
n
), whenever
i
°
=
j
(ii) for each integer
k
there corresponds an
a
i
such that
k
≡
a
i
(mod
n
).
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 Spring '09
 W.Alabama
 Division

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