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Definition 3.1 (Complete Set of Residues)
A
complete set of residues modulo

is a subset

of size

whose reductions modulo

are distinct. In other words,
a complete set of residues is a choice of representive for each
equivalence class in

.
Some examples:
is a complete set of residues modulo
.
When
, a complete set of residues is
Lemma 3.2
If

is a complete set of residues modulo

and

with

, then

is also a complete set of residues.
Proof.
If

with

, then
Proposition
2.1 implies that

.
Because

is a complete set of residues, this implies
that

. Thus the elements of

have distinct reductions modulo

.
It follows, since

, that

is a
complete set of residues modulo

.
Definition 3.3 (Linear Congruence)
A
linear congruence is an equation of the form
Proposition 3.4
If

, then the equation
must have a solution.
Proof.
Let

be a complete set of residues modulo

(for
example,

).
Then by Lemma
3.2,

is also a complete set of residues.
Thus there is an element

such
that

, 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