atfatstott.IT T.EE#T

IMD-12MAR.IO#

Congruent ^{module M-}

-

① ^a _y ^{(mod m}) ^{Se e solo se a e} _y ^damn

hostess rest nella ^division form ^.

② ^{G Sono} esattan.eu#mcLassrdrequimlewte

(ⁱⁿ gusto ^{caso class I} congruent ) ^nella relative ^di

congruous module (^late _quote ^{i rest :} _0,112

, ^... ,m^-t) ^. Insreme delle classe di eguisbnsa ^{G- conquered}

Nome ^, Z/mz

Se indicter are on uwm Sb^{arm la Classe du} _unguents

relates al est ^r_, abbraeeo

← RAPPRESENTED^ONE

74mg ⁼ { ^{E ,T ,-2 ,. - , F- I] CANON, CA}

Es ^. 7462=15/5,775,53

OSI ^{Siam In close tr} _congruous ^{di 7 £ lastsan}

della classe ^d _unguents ^{do I} ( _7=-1 ^{hired 67}

in quant ^{drum hostess mdl,} poker sostrturiee I amI

0 on quaker ^attire

numero adult

.

Un esemp.ru ^do ropy^{mentorme di} Zlmz alternative ^{i i}

2/72 ^{=L -5}₁₁ ^,-2" ^,-5,511 ^{, T , 5,53}

I _, -5,8

(^In gusto ^{caso Sr cesaro anche moment} negatvw ^'_{, yes}^{-0 Saw} pin _tical ^{in v.lore assotntol,}

- _

⇐ ^In 721mg ^so possum ^{fare operation t e a ,}

|SommaI

5,5 ^E 74mg ^→ _vogw^- defense ^Jety Defy Icty ⁼ arty

( _La Sonne delle Classe ^di congruence ^{e- to} classed _congruent

della soueucah

Bisogmnossermre ^che finance ^{tutto tem} (^si trek ^d ane

buon^- defwsimel^.

Are brsogna rerifreau ^{che andre con sai Huo Irene di oben}

hostess wsuetato ^, ET EIN

Se _Je - I ^e 5=5^{' alba 5-5=5' TT}

(^{can nod 6 ) x-I} _y=z x'Et _y't It-2=775 )

VERIFICHIAMO-I.IN^{' und the che se d' (mod m)}

Ori od eat km (keek)

simiemmk 5=5^' _y^' =3 ^{-1hm (the 74}

Akon _x't y

'

a set km + ythm ^{= Cxty) -1 Chthon}

⇐^e-_y^' ₎ ^- _Cxty) ^{I un} mnltgbdr ^on

OE x'_ty^{' ⇐ at} _y ^Cmdm) xIy^' I^{= xtg}

1-ABELLA DELLA SOMMA ^{MODULO 6}

-

-I I ⁵ 55 ^{E T}

-

J 5 I 5 E T I

atfatstott.IT T.EE#T

[email protected]

( lepidolite ^{teller class e-} agnohallaclassedeljsrdolto^?

VER# ICA DELLA BUONA DEFIN^' 210N

?^y j×y^,

Se Je -_I^' e 5=51 ^allow IT ^{= I' -5'}

a' _ex-1km _y^{' =} ythm

N' _y^{' =} ay ^t mckytxh ^{t knhm) x'}_y^{' oxy (mod -m)}

- I

pulsar ^dm FYI ^{=Fy .}

TABELLA PER ^m=6

J 'T I ⁵ I 5

-

E o ^{o o s o E}

-

¥÷t¥¥¥÷¥#

-

Proprietor ^delle _operation^.

+ ^x -0

ASSOCIATNA ST ST

COMMUTATVA ST SI^'

ELNEUTDO ST (8) ^{ST' ( T )}

oPPoST0/1N^{VERSO ST} of ^{do I = -5 SOLO IN ALCON )}

hell^'auto cop^{w CASI} ofpost ^{III 52=4}

-

Dinoto

, ^a blob dresenfro ^{, he} proprietor association perk ^some,

cire (^IL#5)^{+I = Et ( Dta )}

I ^{" l "}

t t

arty ^{tI It} ytz

H ^H

- -

t ^{⇐ Rt9}

✓AUE ^LA PROT^. ASI^. PER^{GLI INTER'}

-

PROB.lt#-Determinarequalielementra.c-Z/mq

fsossredono ^{an inverse} per ^{la moltoguar nee -}

-

Dato _{a- E} 721mg , voghio _copier ^{se esiste a- C-} 74mg the che I. I T

.

I. ^{I -}T ^⇐ ate ⁼ T ⇐ axe I Cmodm)

def^{.I congruent-}

m last ^{- I ax- l =} _{my fer} ^{quick yen}

Ase ^- my ^{= I} CEQ.DK#ANTEAD1PGRADo) Mi interns- quando ^{si n'solve i ⇐ Ca,m) It}

|

Caim)=L

PROBLEMS ^{AVANTI SONO QUEST I} &_infuse^nudism) ( ^solution pain ^avanti)

CASoPARtkdARE ^M =p ^manner fswmo ^-

E , T

, I_,^{- --} _, pI ^{at co, 7=7}

(^a, p ^{7=1 ⇐, I to}

( aight}

2°CASoPARTIGLARt# ^me ft ^{f form , ↳7 .}

(_aim ) ⁼ I , f ^, p?^-- sph ^{( divisor} tip⁾

Quinto Cain) ^{=L ⇐} pfa

Conclusive : Ca_, m)^{=L eeabto net case in an} f) ^{a a=pb}

O , 1,2^{. .- - - - ,} ¥-1 ^{( 57}

0 , to ^,2P, sp^{. . .} fpktdpcpt-i.pl

-

Quest ^{runner some} dolls form _p^-b (by^?

Con O ^⇐ be ph^{' r. DNTOTAE SONO} pm

TOLTI _QUEST)

, RESTAINO ^{QUELU PRINT ON} f ^c

TUTTI ^- QUEST ^{X '}

Pk ^- pm ^{. (}generation^{-I can bet)-}

-

EQUAZDNI ^{CON CONGRUENCE} nv②

-

SOMMA Ita =I

-

Seh ^{E =} It _off^{-sttra =} -5 ^-I =

(^{m -} 7 ^Et -5=5 I -5-5--5=51 ^.

MOLTIPLICAZIONE I.Je=I

-

(lnnanzrtutto non E chiaro ^a priori ^{se questa} equaxnuskrisohh.ie^?

a-_in ^- 5 AT =I _⇐ axe b ^Cmodm)

m last^-b _ax^-b =

my per gquabhe ^YEN

C ^ak- my ^-3 Lea^. drofontadr ^# grodd^.

QUESTA EQUAZIONE E

' RISOLUBILE D= (^{A, m}₎ / ^{b .}

← ^x

SE ST ^{SE NO}

EIJLISOWBILE NONI RISOEUBILE

ESETIPIO ^NUMERICO 1507J

-

MODULO ④ ^52.5=5_{I 7}^-

12 see b (model)

E^' RISOLVBIVE ⇐ E^' RISOWBILE 122C ^{- 21} _y ^-b

⇐7 3=42,21 ^{) lb}

Se b= ₁₃ F

Se b -_ 15 ANDIAMO AVANTI

122 ^- 219=15

DNIDO PER 3 _: 42C ^- Fye ⁵

EQAUSKIARIA^: da ^-7- yet ( Algorithm ^dendrite)

TROVO AD occtllo LA SOLUZLONE 2=2 yet ^. 6% ₎

Sol-PARTI LARE DI ⑦ i K= 10 yes

Soh GENERALE i

{ ²⁼¹⁰_y= ^{5 -14k+7k Present}_{2L=3 yet}^ke-I

RITORNANDO AL PROBLEMS ^INIZIALE _, ^{SIANO IN#RESAT A} DETERMINATE QUAN ^{SONO GLI x "}BUON^{'T" .}

see 10 +7k _,

areato Cmod 77

( attwmentiaanohe see 3 (mod _'t) .

-

|lNGtNERAU

aa=b ^{(mod m)}

deca_,m> lb ^0k

are ^- my ^{-_ b} /. .

'

,-

,

- X ^dela,m)fb ^{No → FINE}

[

a. ⁼ da_, ^m=dm _, ^b=db ,

SEMPUFICO and^- _{M ,}y ⁼ by Ca_, ,m,)=L

. EQ. AUSIUARIA apc ^{- M ,} y ^=L

EUCLIDE (^{o "}A occtho^" ) ^deed _{, 9=9}^, SOL PARAG ART AUGUSTIN MOLTIPLIG _PED b

, I =bpq=2• y=4y,=Yo ^{SOL- PART}

DELL^' EQ^,

ORIGINAPVASOL^- _GEN.

{

IN TERMINI DI se ,

|xexoCmodm#I

QUINDI LA ^{Sol. E' UNA CLASSE DI CONGRUENZA} mod rn ,

ax =3 ( not ^m) ^{→ a},x,-=b_, l _→ ^- a=xoCmk

-

=

Ese^{. see a} fate ^{de una congruence}

sense f. ^{Hori comuni , 'U module dell' epuatvrw -}

quell^{o della sslhtrme coincides}

GI ^{Prime overawe 122=15 ( nod 21)}

e atsbauoottenuto ₄ a ^E 5 Cued ₇₇

Calhoun_12N dowager ³

tuttiehw

-1219=15

II

4.a ^- 79=5

-

APPLICATION^{E : PROVE DI} DIVISIBILITY . -

SW Huron dear mode: n C- IN

n = an^- 10kt _ah

, 10£'t .- + a

,-10 tao

n ⁼ 3276 ⁼ 3+1,0? ^{t 2-102}₁₁ ^{t 7. to}₁₁ ^{t 6}

1000 too 10

① ^{Diwali wk} _per ^{2 o} _yer ^{5 i}

bask guerdon ^{e' ultima} Wfm ^{i ao}

lmfttird.to?eo3.--..sonotuAidiwsr3ihryor2eper5

21h (o 5)^a) ^{⇐ zlao Co 5)a)}

h = Ca₌^{, t --- ta,lo) too}

for^. per ^2e per^J

db⑦ ^{Dinis broke} yer ^{4,25 , 8,125 , 16,625}

2252 I ⁵³ 2454

Rees^. saguaro ^{8 , 125}

⑦ ^{Carlotta tag to)} tq10tajl

8 In (^{o 1251N}₎ ⇐ 21 aztota^{, tax} Coffin ^{1251 --- I}

nd 34126¥ ^{②In-⇐ 81262}

|PRoVADELN0VET

-

OI ^{10=-1 ( mod} 9)

102=10.10 et^- I (m19)

103 ₁ Cured 9)

token= cured 91

n = Az totake ,to"

t^-- ta'^' 10+90

CONGRUENT

µ ^{M '"} model⁹

I 1 I

n E Eutaw,-1 - tapas Curd 97

L

SOMMA DELVE C)FREDI ^{n n}

9) ^{n ⇐} 91 ^{* sauna delle} _afe ^do n ( proud ^{Iwaki Hai )}

(UTI ⁴²²⁰ DELLA PROVA ^{DEL NOVE i} See k ^{e- Il}

risotto della uvtsjhrasme fu ^{m e n ( Kann)}

allow in portolan ^{ke mn (mod 9)}

L

som delle ofe ^{( Sauer- delle ofe dm)}

w k . (^Souris delle_arte din )

COND)ZONE JVECESSARIA

, NON SUFFICIENT.

-

fSTESSAOSAPERLADNKIBIUTAIPER3n.IO#(mod3t

-

CRITERION _{DIVIS) B)LITA}^' _{PER 11} :

/10=-1CmsTy

102€ -11 (mod 11) 103=-1 ^(model) _, 104€71 ^(model)

Segni attend ^.

n = an^. 10kt _aw, 10kt .- ta_,^{. too}

CONGRUENZA module 11

ME an Team, ^. C-IF^{'t -- i -ai tao (modem}

= (_aotaztqx ^{- -} J ^{- Caritastaste - )}

SOMMA ALTERNATA _DELVE OIFFE ^.

Mln ^⇐, 111 ^{sown . alternate delle} sue arte

EI ^{n = 3742} (3+4)-(7+2)=7-9 ^No

n = 3146 (3-147-(1+6)=0

OSE FINALE K numen 7 ^{e- "}SGHEMBO^" WHA ^{Ik base 10}

- per ⁷

e qintar ^{non c' e-} unaitenlodrdrwsohibhi ^cos seuyloa

us and he ofedeuuin^-

40=3 Crud 77 102=2 _Cinta 103=6 ^Cued77 104=4 (wI7) 105=5 ^CHA 166 ^{Luo D- 7}

- - SEQVEJNZA DIFFICILE _{DA TIE} NORRZARE ^-