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LECTURE 19: THE INTEGERS MODULO
n
(
§
8.6)
Appreciation is a wonderful thing: It makes what is excellent in others belong to us as well. Voltaire
(1694  1778)
1.
Tables
Recall that the relation on
Z
given by
a
≡
b
(mod
n
) is an equivalence relation. The equivalence
classes are [0]
,
[1]
,...,
[
n

1]. We will call the set of these equivalence classes the
integers modulo
n
,
written
Z
n
.
For example
Z
3
=
{
[0]
,
[1]
,
[2]
}
. (Be careful [1] means diﬀerent things for diﬀerent
n
’s.)
The elements of
Z
3
are sets, but we still want to think about them as integers, and be able to add
them and multiply them. But what would that mean? Addition would be taking two equivalence classes
and replacing them with a new equivalence class, by some rule. The rules we
want
to use are:
[
a
] + [
b
] = [
a
+
b
]
[
a
][
b
] = [
ab
]
.
These seem nice. Let’s see what happens in
Z
3
. We can form the addition table and the multiplication
table. (Done in class.)
There is a problem here. Why does addition and multiplication work? For example, when adding
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 Spring '11
 Smith
 Integers

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