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Definition 3.1 (Complete Set of Residues)
A
complete set of residues modulo
![$ n$](img4.png)
is a subset
![$ R\subset\mathbb{Z}$](img46.png)
of size
![$ n$](img4.png)
whose reductions modulo
![$ n$](img4.png)
are distinct. In other words,
a complete set of residues is a choice of representive for each
equivalence class in
![$ \mathbb{Z}/n\mathbb{Z}$](img3.png)
.
Some examples:
is a complete set of residues modulo
.
When
, a complete set of residues is
Lemma 3.2
If
![$ R$](img50.png)
is a complete set of residues modulo
![$ n$](img4.png)
and
![$ a\in\mathbb{Z}$](img51.png)
with
![$ \gcd(a,n)=1$](img52.png)
, then
![$ aR = \{ax : x \in R\}$](img53.png)
is also a complete set of residues.
Proof.
If
![$ ax\equiv ax'\pmod{n}$](img54.png)
with
![$ x, x'\in R$](img55.png)
, then
Proposition
2.1 implies that
![$ x\equiv {}x'\pmod{n}$](img56.png)
.
Because
![$ R$](img50.png)
is a complete set of residues, this implies
that
![$ x=x'$](img57.png)
. Thus the elements of
![$ aR$](img58.png)
have distinct reductions modulo
![$ n$](img4.png)
.
It follows, since
![$ \char93 aR=n$](img59.png)
, that
![$ aR$](img58.png)
is a
complete set of residues modulo
![$ n$](img4.png)
.
Definition 3.3 (Linear Congruence)
A
linear congruence is an equation of the form
Proposition 3.4
If
![$ \gcd(a,n)=1$](img52.png)
, then the equation
must have a solution.
Proof.
Let
![$ R$](img50.png)
be a complete set of residues modulo
![$ n$](img4.png)
(for
example,
![$ R=\{0,1,\ldots,n-1\}$](img62.png)
).
Then by Lemma
3.2,
![$ aR$](img58.png)
is also a complete set of residues.
Thus there is an element
![$ ax\in aR$](img63.png)
such
that
![$ ax\equiv b\pmod{n}$](img64.png)
, which proves
the proposition.
The point in the proof is that left multiplication by
defines
a map
, which must be surjective
because
is finite.
Illustration:
Set
.
Then
so
.
Warning:
Note that the equation
might have a solution even if
. To construct
such examples, let
be any divisor of
,
any number, and set
.
For example,
has a solution!
Next: Fermat's Little Theorem
Up: Lecture 5: Congruences
Previous: Rules for Divisibility
William A Stein
2001-09-20