Math 100
Congruences and Divisibility
Martin H. Weissman
1.
Congruence as an equivalence relation
Recall the deﬁnition of “congruence mod
n
”:
Deﬁnition 1.
Suppose that
a
and
b
are integers, and
n
is a positive integer. We say “
a
is congruent to
b
,
mod
n
”, and we write
a
≡
b
(mod
n
) if there exists
m
∈
Z
such that
a

b
=
mn
.
Other interpretations are the following:
•
a
is congruent to
b
, mod
n
, if the diﬀerence
a

b
is a multiple of
n
.
•
a
is congruent to
b
, mod
n
, if the remainder after dividing
a
by
n
equals the remainder after dividing
b
by
n
.
•
a
is congruent to
b
, mod
n
, if
a
is equal to
b
plus a multiple of
n
.
We begin with the following observation:
Theorem 2.
Let
n
be a positive integer. Congruence mod
n
is an equivalence relation on
Z
.
Proof.
Congruence mod
n
is reﬂexive, since if
a
∈
Z
, then
a

a
= 0
·
n
, and hence
a
≡
a
(mod
n
).
To prove that congruence mod
n
is symmetric, suppose that
a,b
∈
Z
, and
a
≡
b
(mod
n
). Then
a

b
=
mn
,
for some
m
∈
Z
. Therefore,
b

a
= (

m
)
n
. Hence
b
≡
a
(mod
n
). Hence, congruence mod
n
is a symmetric
relation.
To prove that congruence mod
n
is transitive, suppose that
a,b,c
∈
Z
, and
a
≡
b
(mod
n
) and
b
≡
c
(mod
n
).
Then
a

b
=
mn
, for some
m
∈
Z
, and
b

c
=
kn
, for some
k
∈
Z
. Therefore,
a

c
= (
a

b
) + (
b

c
) = (
m
+
k
)
n.
Hence
a
≡
c
(mod
n
). Therefore, congruence mod
n
is a transitive relation.
/
Since congruence mod
n
is an equivalence relation on
Z
, it partitions
Z
into equivalence classes, which are
usually called “congruence classes (mod
n
)”.
Proposition 3.