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SET OF INTEGERS MODULO n
1.4.1Definition. Let and
0 beintegers.Theset
ofallintegerswhichhavethesameremainderas
whendividedby iscalledthecongruenceclassofa
modulo ,andisdenotedby
,where
Section 1.4
∈ ≡ mod
Integers Modulo n
Thecollectionofallcongruenceclassesmodulo is
calledthesetofintegersmodulo ,andisdenotedby
.
Anelementof
iscalledarepresentativeofthe
congruenceclass.
ADDITION AN MULTIPLICATION OF CONGRUENCE CLASSES
1.4.2Proposition. Let beapositiveinteger,
andlet , beanyintegers.Thentheaddition
andmultiplicationofcongruenceclassesare
well‐defined:
,
,
,
in
,thefollowinglawshold.
Associativity
⋅
If
,
then
.
∈
and
0 ,
iscalledtheadditiveinverse of
⋅
ARITHMETIC WITH CONGRUENCES
Foranyelements
ADDITIVE INVERSE
⋅
⋅
⋅
A DIVISOR OF ZERO
1.4.3Definition. If
belongsto ,and
⋅
0 forsomenonzerocongruence
class
,then
iscalledadivisorofzero.
Commutativity
⋅
Distributivity
Identities
⋅
⋅
⋅
⋅
0
⋅ 1
AdditiveInverses
0
1
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MULTIPLICATIVE INVERSES
1.4.4Definition. If
belongsto ,and
⋅
1 ,then
iscalleda
multiplicativeinverse of
andisdenoted
by
.
isaninvertible
Inthiscase,wesaythat
elementof ,or isaunit of .
A COROLLARY
(1) Thenumber isprime.
hasnodivisorsofzeroexcept 0 .
(3) Everynonzeroelementof
multiplicativeinverse.
hasa
A FORMULA FOR THE EULER
‐FUNCTION
1
1
1
⋯ 1
(a) Thecongruenceclass
hasa
multiplicativeinversein ifandonlyif
gcd ,
1.
(b) Anynonzeroelementof eitherhasa
multiplicativeinverseorisadivisorof
zero.
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1.4.7Definition. Let beapositiveinteger.
Thenumberofpositiveintegerslessthanor
equalto whicharerelativelyprimetop
willbedenotedby
.Thisfunctionis
calledEuler’s ‐function,orthetotient
function.
THE SET OF UNITS
1.4.8Proposition. Iftheprimefactorization
of is
⋯
,where
0 for
1
,then
1
1.4.5Proposition. Let beapositiveinteger.
EULER’S ‐FUNCTION
1.4.6Corollary. Thefollowingconditionson
themodulus
0 areequivalent.
(2)
DIVISORS OF ZERO AND MULTIPLICATIVE INVERSES
.
1.4.9Definition. Thesetofunitsof ,the
congruenceclasses
suchthatgcd ,
1,willbedenotedby .
1.4.10Proposition. Theset ofunitsof
isclosedundermultiplication.
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EULER’S THEOREM
1.4.11Theorem(Euler). Ifgcd
then
≡ 1 mod .
,
FERMAT’S LITTLE THEOREM
1,
ThefollowingcorollaryofEuler’sTheoremis
knownas“Fermat’sLittleTheorem.”
1.4.12Corollary(Fermat). If isaprime
number,thenforanyinteger wehave
≡ mod .
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