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~~auo~~naanuwun~oa~n~iurrda 16uri '~1Q raaAOaui3n ~ u i ~ w ~ 4 &nim rlcnougau unm.aiui,

UCI'TUIIU~,

a ~ y +uardmiuy?

(2)

r.lrmao~ii~?uduuanA~'ounii 10 I$rrrri ~aaddaui~nriru 1,2,3,4,5, 6,7, 8 rra:

(3)

~aauoarauntYnIu~iunn 16uri ran~iau13m~u 0,

(4)

r a m o o a ~ s u a e m t Y n ~ ~ ~ 18 u wl6uri ~ ~ ~ ~ramdOaui$nr~u o~ 18, 36, 54, 72 us:

(5)

raaddcnouri?sraauoa4iuauriu~~~~:~~muoa1iuau~iu~ ramifi4iuauaui3n 2 Ga us:

I, 2,3,4 us:

u~: B rlluraaun.s4iuauriad aui%n&aaa~fluraa$1"ilcs" A ~Suraauo~4iu?urRud u h C r~uram~dsmnu&uramu0a4i~aur~u~~s:raauoa4iuaur~u~

&I&

A E C , B ~ C u a : 3 ~ B u 3d @ C

(3)

..., k a i n u } LII~IOJ~IU~ULL~U 1 8 4 100 ~ d u u ~ n u h{ u I2,,3, ..., 100) r~a~o~4iuau~cbw7rt ~ 4 u u ~ n u 6 a u{2) 3, 5, 7, 11, 13, ...}

(4)

L=LfR%Q49R

(5)

{(I, 2)) (2, 71, (3, 121, ...I ~ r r m o ~ d i u a u ~ i i u d r i o u3n i duuunu6au i {..., -1, 0, 1,2,3}

(6)

~rnuo~4iuau~iiudnisiYau 5 ad&

(I) (2)

a'lraaosb#ou ~ i u u w u d i a u {unnnu, guniGuG, i w n u ,

(x, y) d ~ ~ f l ~ l j ~y d= ~52~ ~3 ~L ~f ElIx l ~fl~~110~cU ~ L~UUWU~~LI

duuunu6au

{

-10 -5, 0, 5

had16 2.5

in' A = {I,3 , 5 , 7 , 9 ) uat B P~~~U?IQU%I 8.1 ~3128d7A =

Ri A

= {X

~Ju4iuauildo~~tnii~ o 3.1 10)

qzndial6ii ma B ~ l l ~ ~ r n l a %A' ~ ~ ~ ~ o ~ ~ ~ a

61 A E B uar A

A c B atiaunwa A i7~lluZu~~mufi~m B ~iuu~~nu6au

f i ~ q aA !zj~Ju&baaaas~aa B

arr4uunnuGau A

nqw&wi 2.2

h o d i d 2.7

IG' A

uar B ~du~qmlm q

B iiia~dn A 2 B ua: B c A

~ d a ~ o i n {u~nzna,&lo) { u t d a ~ UIRLIIO, , n h u , &lo, ~ r l t 5 ~ ) un: {uzrja~,uznrna, tdau, gu} ) {YV,u z d a ~ u:n:no, , n h u , $u) &$u {uznzno, &lo) { ~ y uu, t d a ~ u:azno, , n A u , riulo)

63aei1s 2.13

1%A

= { a , b, c,

6hfu A

..., z) ua:: B = {n, TI,n, ..., a)

uw:: B ~!Ju~.nmlaidhufao~

otb~ul~41 A nB =

{Jar, n n u , ZIL~U, dua, mior, u m , UWJ) uat A = {umr, cntar, &~fiu) otld' A1 = {dad, n n u , ~iiua,u m )

hi'

WJUU

uat

= (LHI?,

a7hd)

AUK = 0

miawiIm} La:: A = {miayn, aiaqnf, Ian} F ~ Z L M * M ~A ~ ' ~C

LLAZ

A1 = {miaknia, miawqv7mii, a i a ~ m imia;~rC~, , m i a ~ u d ~ miaw~Im} u,

(A1)' = {miaya, miaqni, Ian) = A

'mars

AUB

A-B

B-A

Cnfiuu 55 nu ~slniw<snqu Cfl6uu 50 Flu N @ ~ ~ I M I % ~

I-

3. o~dsudl:~ondo!d~mu'I%<~fna&vosrqm 3.1

1 b.hfiuaui%nvos~cnmA

3.2

'Imnbsuoubguaui%nvosrawB

3.3

~ ~ ~ i ~ ~ ~ a r t 9 ~ u i u ~ f i uC ~ u i Q n u o ~ ~ a ~

3.4

ram D dcrcnou6au~ui~n~s~1um~uram A E ~ ~ c n ~ u ~ ~ u ~ u i ~ n A ! ~ j i i i iAu uuo ~ ~:Ba m ram F dcnou~auau~Qn#\ioi'Iuram A udoi'Iuram B uar C

3.5 3.6

4. ssun~an~ii~ram6d~uu~da\d8 4.1

4.2

A uac B r~uramds B rflu<ub%mufiuo~A

4.3

am Ad-tiiuauaui%nrZ~upd

4.4

A uwr B ~ f i ~ b d n 9 # ~ ~ i ~ a ~ H ~ i ~ n b ~ i i < ~

4.5

A, B uac

MA:

L~~ML~~~\~GHUI$T~$?%.JKU

b~u~am~~~mur~mdwu7~n~~ij~uu~iidis~u

5. ad~~uurwi~?n%.aags~~gado!d6 5.1 {tud,0) 5.2 (0,1,2,2,5,0) 5.3 (00,01,10,11) 5.4 {4nwod, utarns, fu) 5.5 ~(nasl o~~u~uGdai$~!xiilwst

8. odvighqw~mhtd7 sio!dd nqwiun 2.5 $a (ii), (iv) b~at (vi) 8.1 8.2 nqwaun 2.6 $8 (ii), (iii), (v), (vi), (viii), (x), (xi), (xiz), (m),(mi)uat (zvii) 8.3 nqwauw 2.10 nqktzun 2.11 $a (ii) 8.4

9. Ihmun~nii~.na A uar B ~8u~amln 9

9.1 9.2

nadau~dsunsu~dnrii~iud~do!dd

L~ML-I.~ALY~I~~U

f i n i m i j i ~ g aA

LQ:

finifmi4i~ga A

L ~ V ; L ~ O ~ ~ ~ B I S IBO JMjo~9.m L~~ B

L ~ M ; L ~ O A ~ I ~ SA I B J L ~

9.3

k t i f ~ b i j im L~~UR~JI%~"PIQSL'II~YI~O!~~

9.4

mieradi~.uos~.nm A

9.5

~I~I~~MIUI%~SI~~LWIL~O%% A ~(Go S I ~ JnL(~P~( A ) ) )

LLW:

~ ~ i ~ ~ ~ w ' ~ h ~ ~ ~ i ~ n l 3 ~~1 a8i ~~~ u ~tu r ' U7 h~t ~~i i Y~ ~~i Mia n~' u~; nnaiu~uw'uhrvlii~ ~ ~ ~ i 3 ~ f i ~ J

~lu~u~iiuaoJ~luau6b1wuan~i3~au 5 aJ&

du6u

ddo~dor~oi~ruin~~~~uu~%@'Lu,fd~~uu~o~

au~%ndiirniu~u~udn'ulu~dda"uGu

CI aI LLRZ W b LJUAUI~~I~ " 7 ~ u ~ m ~ ~ n i r ~ i i au uLLRL ~ ub i ~ ~.SLGUU n (a,b) bun ii dhrill 6un a ii a u d n h ~ huarnun b ii a u d n & w t ro~rnsduui7naiuh~yda~a"udu ~ w n r d u(a,b) r (b, a)

ao.rd5udu

da~oiniYudu

U ~ G U I3.N2

I$ A

LLao B 6I~1,qmIaq

A x B Bo ~aausr~ju$u(a, b) .L a

uunn

63eei19 3.1

<u a

P . I ~ ~ M R I % P ~ L % ~AU " Laatwa U ~ J LB~ ~~ ~ ~ L ~ U U L L W U ~ ~ U

a E A wr b E B

A x R = I ( n h\ I n F A uaz h F Rl

A = {I, 2, 3) kar B = { n , 3 )

B x A

~ n r ~ n i ~ d i n ' d m u i a r i ~ a s " ~ ~ r n r iaeau. i,m~~i~a~~~~

k a d ? 3.6 ~

A = {qai, uis, $, %, G%f,u q % ,!a@)

ne:: r ~3unaiuguGud u~iluqiGiiYwuu A u

r = {(a, b) E

x A

illuqiIIYi~ b)

~ 4 %($, w"%4)E r ~sJiunaisJji y%!iilau~i~~uw"%"a L: .I

h~?unaia~GuW'UdT %a9 u i ~ ~ 1 ~ ~ ~ ~ ~ ~ 1 a i u ~ i ~ a(a, a b)n ~ E ~r ih~ a~ Q a rubu du unu 5-1 r ~flunaiuCuGu6uGaunjin

6 ? e d i ~3.7

I$ N I$ uar

~ U L % ~ Y ~ J ~ I W 9:~~~luumu ~ ~ B ' ~ J (a,b)

uwu~qiwued4iuaut~uuan

;

E r $ 3 ~a < b ~i1uGu

= {(x, y) E

r2 = {(x,y) E

N x N 1x <

B::!&~Irl u r. = {(x, y) E

y)'

M 1 x = y)

NIx 5

ii1unai~CuGuduu N y)

L~~M~~IUGU MGU~JU

1%A = LamaoJnu ue:'LM' rl rfIuna~a~$uGubU i ~ n l ~ ' 7 , r2 riluna~uiiuw'ub"~3uuri"unr r3 ~fIun-~~u&w'ud c'ai~5unuu A &&A

A o d i a 3.8

r, = { ( a , b) 1 a p n 4 i b) 5-I = {(b, a) 1 a g;lnii b)

ez\&

= { ( a , b) I b gsnii a}

r2 = { ( a , b) 1 a rfIuuiva~b) r2-' = {(b, a) 1 a ~Suuriqos b) = { ( a , b) 1 b r3uuJaa~ a)

r3 = { ( a , b) 1 a aiiniiu b) r3-I = {(b, a) 1 a aftw{u b) = { ( a , b) 1 b aftw5u a)

xW1y

= x)

WxB1y { ( x , y) E W x B 1 x

= x}

r, = { ( x , y)

= { ( y , x) =

7-2

r;'

EW c)E

= {(x,Y ) E = { ( y , x) E = {(x,Y) E

x x

= y)

IY Y WI

22)

2 22)

Ix 2

2g)

2 1 r3 = { ( x , y) E R x WI Y = -) 3 x+l rL1 = { ( y , x ) E W x Y = -) 3 y+l = { ( x , y) E W X W 1 x = -} 3

Gaadw 3.10

Iw" A

= (1, 2, 3) unr B = {n, a)

Lu= u

= {(I, n), (1, 91, (2, n))

D, = {1,2)

unr R, = {n,a)

~ o i ~ m i l m u u u a r f i ~r-I u~a~

oin u I t m a

{(n, 11, (3, l),.(n, 2)) D,-' = {n, a) La:: R,-' = {I, 2)

r-'=

oi r ~ S u n ~ i u l G u k iAn 1d B or16

D, = R,,

uar

D,,

unt orlb

&adw 3.12

W A un::

{yai, U I ~ R I , @,ns%, *ibS, w$%,Ia??u, n n i i n i )

B = {n 011, n 012, n 013, n 014, n 015, fi 016)

Iw" rl,

r2 aSunaiutYu$uioin

A I d B uar s ~ S u n a i u ~ u $ u h uB

R-IWM~~~U

Gninliuu x ~ i ~ u i aa, i iida~i?a r i n h x m u i a i y wiu (x, y) E r2 n (x,y) E s iidarda ~ou?ais diufiauinliuu?qri y A~ilr$i.ai.rv~~~moin s o ~ t S i n a i ~ ~ ~ i u ~ a ~ n a ir1 ~u~ u T2,w rl " un8r2, rl - r2 uar s o rl ( 5 , y ) ) ~ r ~ iida~:~

1-1 Ldu

hed-la 3.14

= r

ua:

r"+l = F o r

&M?U

72=ror,73=720r

6 9 ? a a l n a i u ~ W i h u{ a , b, c) d a ~ d d ' = { ( a , a ) , (b, b), (c, c ) ) 7-2

= { ( a , b), (b, b), ( b ,

4 , (c, 4 )

n = 1,2,3,

...

r3 = {(a, a), (a,b), (b,b), (b,4, (c,4) ~inuniiuiuodkrl uwr r3 dauuZiaz6au Gs;y1' 3.5(1) uw:: 3.5(3), m?urf?Biu kur

uBi

r2 b.iBj~~%arP?'auLWM::

(a,a) @ r, &$1' 3.5(2)

Z unu~a~wod4iuau~~u rbwr R unub%awos4iuau-6s

@bepials 3.15

fi~1ismicl~iu~uw"u~~a1d~

z x z 1 x MI^ y asga} {(x,Y) E X R I x > Y) {(x, y) E W x R I 2 + $ 5 1)

r, = {(x,y) E 7-2

bba:

r3 =

~ c l n ' r, dauu31n'ou ~m:: x ni3 x

u ' u o (x,I)

yndi x s Z

RJ&

rl yn x E Z

r2 llidauu3a:fiau

bwnz x )I x qlndi x s R

$u$o (3,x) @ r2 yn X E R r3 UBjauu3a::n'au

L W ~ Cl 2

+ l2 =

> 1 kuto (I,I) $ r3

- - . - - ... . ., .-....JRaJWUL- .. .- .-. ... . - .. r rfluaa~ola'srw'p16auyaupbA

61 r iauu2azfiou aoluiar ~~a~si7unoa

~ u %61~ r ~ilunaiuei.wigaqauw A

L I , ~

1. ~ M ? u A La~E~ A, : ( a , a) E r

2. hps?uudaz a, b E A,

&20d19 3.20

81 (a7 b) E r uga ( b , a) E

%tfi r i ~ U ~ 1 ? 7 u ~ a JZ~ u n"1vlum~au ~U ( a , b) E r

fi k E Z

HOL~O

d a-b

= 2k

~BWR~~U%U&#~U;I

~ J U A R Jr ~ ~

?%I

24' a E z

~1~15 a-a

~ q oa - a

= 2(0) ius

oEz

i. a

&&A( a , a) E r UURO r d~uWarYiOu d 2. !w' a, b E Z ria:: ( a , b) E r ed$ii d kEZ n a - b = 2k t

iwmrii b - a = -(a - b) ~ w m r a z ub~- a = -2k

OL(a, a) E r 3.

ki;o

b - a = 2(-k)

L ~ J O-k

r dauuSlauwa3

a, b, c E Z ua: ( a , b), (b, c) E r

b - c = 22

W?O

or16 ( a - b)

+ (b-

Q ?!~:I

d k, K E Z 4 a - b

c) = 2 k + 2 K

a-c

= 2k ~8::

= 2(k+k')

bdo Ic+ICâ&#x201A;ŹZ

6h.h (a, c) E r i h i i o r daurul3diunoa in I . - 3. ~ ~ r1 ~BunaiubW"U$~urulu 6 z

rrJr 6

... , ....

n ( B ) = n MAI? n(A x B) = rnn

.,~rnfiinuanaiu~uw'uBaii.

iwn~hnaiuguwirbin A I d B b~u&~qmaorA x B k t u

~iuaunaiu$uw'uB~inA I d B ~nin'Y 2" & &

oinni3a2"1anaiu$uw'ufi6'YI~iu~~flruldbo~Q L W U ~ I ~ 'A

.rw

mai~$uW'P6libs~rul~u~~auwnwwG~~[w bdu KaoLiiaia!dtb

ma~u$uw'uh~vri1~3~u~wia r i i=~ ~{(t, a ~ i s) E R x R

= 2f

+ t + 1)

rBuna~uii+uw'ud

~~~ii~b%maoaba~i!d$~ b%mao94::u~n7a ~ ~ ~ ~ ~ n ~ ~ & ~ i $ ~ b a ~ i ~ d ~ u ~ ~ ~ t n i ~ ~ b d ~ u u ! d &

n a i u 8 u ~ ~ a " ~ : w i i ~ C n d s u u d ~ 6 ~ ~ 5 ~ m ~ ~ ' ~ 4, u 33~, 2i ,n1,~ ~ o ~ 1bflu~aiu%uw'uht~ii~ 11ai~~tb r ~ a a o ~ ~ ' n f i u u d f i u u m ~' a~ J~E~Si L~' a~ IQ~ " U O J S = ~ U ~ :4L, L3, ~ M2, I , o ar8dbna!6i1 Cnfiuudfiuu , 34 n~amaaibbdn~mcba:!6~t~~m~uu~6bvi~~u~3t6rul~~ua naiu~uw'unb'6uubaoniiqa6.a"~ 5~rulwftuiude!d6 8.

&ae(i19 3.21

fii%ua A = ( 1 , 2, 3 )

LW~:<I

MIL

B = {a, b)

( 1 , a), (1, b) E h u9i a

&aopiis 3.22

(1)

~sfi~1if~~iiinaiu~uw"~~tii~um1~do~d~~~u~;~~6.a'un~o!~b~i

A unu~aarosdasinshamn~~~n1nlu~iu~3~s~um B unu~amuos~ass~na~Bu$iim~nian~a~wwu~iun~

uazIG r ~ l l u n a i u L G u ~ n ~ ~ ~ ^ n ~ 3 a n ~ ~ G n ~ ~ ~ u $ i i ~ 7 ~ n 1 3 n ~ ~ ~ n w u ~ i u czl$ r ~Bunaiu%uWUM$-ainA I d B LLnz r ~BunaiuguGu&~Bu4s6.a'u (2)

r ~Elunaiuriuwica'"&u# ( A ) uocaanau?ad R7' uu~aslaashuauaistiinumbu

( A , R) E r otI&

uwGeriar 3.12

iidardo

A = nR2

LC~~~I~~?U~"P~$~BMAJ~%W

I$A ua: B ~3u~"a~lm q ua: f ~ijuA~6.a'~ A Id B

$ti (x,y) E f

dun y ilri?udmiud$~$uia~~d~8w~t x Qu~ulaey ~3uri1uos&6+u f

6aatii;r~du ( 2 , 4) E f nuiunaiuii f(2) = 4

~nmuianuaas~~65u~muoi~uu~Guiu 3.12 I 6 ~ d u

f = { ( l , a ) , ( 2 , b ) , ( 3 , c ) ) bb~msIAKau f(1) uaz

g =

{(x, y)

Rx R1y =

= a, f(2) = b, f(3) = c

J Zrr~asl$v7au )

g(x) =

JFFZ

hn?u x E B

6aod19 3.23 (1)

fiairainaiuEuGu#

AX> = 9~19~1fl-1),

; x < 1

x - 1 ; 2 ;

f(2) uar f(5)

1 < x < 3 2 2 3

6aogiid 3.24

Ics" A

= (2, 4,6,8,10) bat

Lla:

= (1,3,5,7,9)

= ((2,

I), (4,1),(6731,( 8 > 5 ) ,(10,7)) 9 = ((2, 1) , (4,3),(6,513(8,7), (10,9)) h = {(2,1), (6,3),(10,7)) 0tl&.b41 f : A + B LW%7t D f = A uat R f = {1,3,5,7) C B

bla:

2 B lwnt D, = A uat R, = LCIVAJ~%Vud!a.j~~ufi~6<uin A ‘1d B

g :A

LWMZ

h du$dd$~qin {2,6,lo) d!

D~ * A

Gaadia 3.28

1%A = (2,4,6,8,10) 1%f,g, h : A B nd

bbwr

= (1,3,5,7,9,11)

-t

((2,I), (4,3),(6,517(8,7),(10,9)) 9 = ((2, 11, (4,119(6931,(8,5),(10,711 h = ((2,31, (4,I), (6,7),(8,5),(10,11)) 8Zlg"h f bCIZ h L B U A ~ ~ % % M & Q ~ B M ~ ~ unr g ! a i ~ f i u f i 4 6 % u ~ d ~ d o~~~d3~7 : (2, : I), (4, 1) E g uBi 2 % a h 4 3.29 na o

35917

AX) =

v7oistui&fi%u

bMOJ0lT1

1,-1

!~~LJUY~J~%UM~~~OM~J

f(1) = 0

2-1 LIZ

A-1)

xER

1fiur7~n"%uts&do~i~~ldld #

-1

(3)

f = { ( x , y) I y = 22 - 3 ) ~Su4r6.a"undrdondrllu R

(4)

f-'

I ( y , x) E f )

= { ( x , y)

f = { ( x , y) I y =

2 ) ~Ju&6Cu~dUn$dotldauuR d

s:!r$i

f-I

= { ( x , ~I )( y , x ) ~ nf LO) = { ( x , ~1 )1 = { ( x , y)

n i n i i n i (+-)

= { ( x , y) I x = 2y - 3 ) rSuA46%u:r$i2u

auuiiii f

CI f - I

(3, X I ) , ( R 22) E

L w n z i l f-'

2 0 ) ~d~Su4~6.a"~

~ 9 u ~ r 6 sruanrii ~u f ~ilu446<undreiotllw

4 , % E A uE+r Y 4

= &f i

= $

( X I , Y ) , ( % , Y )E f

f-'

sSuA46.a"i.b

LWa7tGMM

~ h f uf ~~~446.a'tbndrdond~

q = q

X ~ O )

k a h l 3.32 (1)

~4fiQll~ldll%~Cf~1~~~!~'1~~~~~~'i8~~ ~lYIM@%M' A

= {n, "U,

FI, 4)

B = (1, 2, 3, 4)

r j o n ~ d r i < u f : A --+ B h u A

An) = 1, A"u) = 2, An) = 2 bag f(d) = 4 ~ w n r a z & f ~ J u v 7 ~ 6 % u r r ~ ~ d o n & aAi n"klkafi~ B txffo A B

UPL

1s f,g : N -t N %K!dijl

fo g

fli~uah~ Ax) = x URL

Mldl!k

UP:

g(x) = x - 1

TslUd

( f o g)(x) = f(g(x)) = A x - 1 ) = ( x - 1 )

( g o f ) ( x ) = g(jyx)) = g ( x + l ) = ( x + l ) - 1

= x = x

2. oinnqwijun 3.2 uarun$uiu 3.18 or!$iii

Lbar f-' o f MI~I!& i n 1 a E A, 6 E B

f o f-'

6irliwualn' f-'(6) = a d o flu) = b h: a5w ( f o f - ' > ( b > = (Kf-'(b)) = Aa) = b uaz ( f - ' o f ) ( a ) = (f-'(flu)) = f-'(b) = a &+Ao f o f - ' = iA LLar f - ' o f = iB ria i, uar i, r i l u 4 ~ 6 % u r a n ~ n ~Ad ~uar u ~B~ ~m i u i i h (f-y = f w+aniiatn<u~dsi7 &.

3.34

rli~ualn'flx) = 1 + 85'191

+ g,

3161 91n'I~ln68zl$ii1 D~

- g,

fg bLag

f 9

= D ~ -=~ D~~ = D~n D~ = { x 1 x 2 2 ) LEI::

DfIg = { x 1 x 2 2 uwz x o

bLar g(x) = x - 3

f mrii!$i f + g, f - g, fg uat 9 (f+g)(x) = x+ ( f - g)(x) =

m-2

JX - x + 4

(fg)(x) = x+(x-3)&3

-3

I ~ r ~ n i ~ 6 i i i n l n u ~ ~ i ~ m a " ~ ~ 1 z ~ r iaoau. i,~~ia~f~aii,

r- Y &9iMu Ax) = A-X) szfiundsn.auu41 %n"Sp1$ Caoliisdu 2, 2, 9,..., c o s x 2. $In%nndmsbfi%u f Aauu~mm~u~nfinii,~ s:\Caa~nia y = Ax) c)ua: -y = A-x) 3 a- d ~si9Mu AX) = -A-X) szfi~n4sn.a~u37 gdnaer~ Caoliis~du X , 2, 2, ..., s i n x 6 . -

1 .o i n n i a ~~un.nd~osaunim i

mui~nt%7~1d~ u~uni~~4uunndvns1unim y = Ax) + C, y = fix) - c, y = fiz+ c), y = A x - c) do c ~0udilntaiuan n7mdidi~sihe uiuannioausin Ax) ~:duasioni~duunndbauni~c)~ounnd~o~auni3 y = fix) lunisvraiun'Yunu y iauni~0%7fii~s6a c uiuann~oausinGaudm3ijmz x ozG~a donim~iuunnd~au 2 mdounndvotaum.r y = fix) lunisuu~un*llmux ~duRaoliitdn\du y = Ax)

is41 *ildusiunirduunnd&unnd"uo~y = 1x1 riou nnvldKa'n~ni~~z~iluni3~~ou~uni~~uiun"~~~nu x 3.21

m *

v-l

-21

\\/ 1

...' y = lx+1

3 8 i i d u ~ i ~ m 3 d ~ ~ n 5 7 d 6 7y~ = n ~14 ~ ~rioouJ n n ~ d ~ o ~ n i . s e z ~ ~ u n i a ~ t o u I u ~ x~ ~ " ~ u i u ~ ~ l barbnu y 6~;d 3.22

~ ~ s ~ n i r d i n 3 n u ~ m ~ n T " ~ ~ w : n ~waau. ~~i~aT"~aG's

1. s~~!uuauiQn.lra~n1iuguGuh6 r 0in A = { o , l , 2 , 3 , 4 ) IdGa B = { o , l , 2 , 3 )

lhnu~buk l ~ ? ua E A,

b E B, ( a ) b) E r

~ida~da

1.1

a = b

1.2

a+b=4

1.4

a1 b

1.5

w.3.a~.us4 a

r LWZ

bifl~~?laJ~.~056&7Mtb'+T~R~

= {(I, 2)) (1) 3)) (2) 3), (2) 41, (3) 1))

s = ((2, I), (3711, (3, 2)) (4, 2)) QJHI

so r

uw:: b

1.3

a > b

1.6

n.7.u. Y

~ Ja

ua:: b

8 . rii~umA =

a ~ l o i m m i i ir ~JunaiuairGuhut(~llu A niob ri~oflinum

8.1

(a)b)r(c,d) ikio~6o a + d =

8.2

( a , b) r ( c , d) ~ d o ~ d e

8.3

(a, b) r ( c , d) fido~do ad = be

-4

b+c =

9. o ~ ~ o i . r m i i i ~ a ~ I m e i o I ~ E d~ o ~l h un f iu~ m ~ ~~u~ D~ ~J u~ u~~L 9.1

( ( 5 Y) I Y =

9.3 9.3

( ( 5 Y)

+ 3)

I = 77) { ( x )Y ) l Y < XI

~ ~ L S U ~ ~ U ~ M ~ ~ J

9.2

{ ( x , y) I y = 2

9.4

( ( 5 3)

= Y

+ 3x - 2 ) + 1)

l o . ~ 9 f i 8 7 ~ ~ ~ 6 7 ~ i ~ J 6 . a " ~ d ~ 1 d ~ L 81~7fl~ J 6 . ao~niA~6.a"uwngu6au "~~~9d0~~~~~~d~

10.1

Ax) = x

10.2

fix) =

2 - 32 + 2

m ~ d ~ n a i u ~ u ~ o a w o a ~ i u a u ' L ~ ~ ~ a d a t ~ i 7 " u o ~4m iu a 0aau~~a a~ ii j ~ i ~ 1 u i n i ~ ~ ~ ~ ~ d f u " u

uardniduurirya*nwd~uh~1~~nu4iuau~~~nii &xaa n q ~ ~ 9 i u a u ~ t l u m i ~ d ~ ~ u ~ ~ ~ n ~ a ~ ~ ~ a4000 f $ ~flu& ~ ~ ~~tlumwidinrn~iuariu~~ uui~iuuinni 6c.uu3~osBiuau~~a.1 9 9 4 0 41uau

..., -3,-2,-1,0,1,2,3 ,... Q i n ~ ~ ~ ~ u l i n d L ~ ~ M a ' ~ ~ l ~ ~ ~ U 3(Mesopotamia ~ ~ ~ t b Y ~ ~ [2000 ~ ~ 1 1 flfiowni~ania]) 1:db~~~ dadnnalu w

tsua~anaoii "concrete problems and onlg one solution given with no attempt to generalize" doui

S r n ~ I n i(Pythagorus ~ [550 flriouni~ania]) ~tlu~Cfbnw~ifsariu1~1~1~3~atnaiu~"Tadi~ 7~ o a

~ n a ~ n i r d 7 ~ ? n u i ~ r i a ~ i ~ a f ~ r i i , n ~ aoara. 1iaaf~aii,"n

d $ o l i "Euclid's Elements" &oerinpdhi7zg 7 msraun$n

G a d i s 4.1

rdu ~ufu.orai%a~lm~n4 i u ~ u r a w i r G r B u

o s ~ o ? a a i j i4 1 20, 7 1 (42), 21 ( 0 rbae 10 18 n%'bsj

aani

1%A

~mu~.ncAaa~Gawia$~~ua~o~ 100 GJI.&I~

A = (-100, -50, -25, -20, -10, - 5 , 4 , -2, -1, 1, 2,4, 5, 10, 20, 25, 50, 100)

da.

Q.3

huau~~uuuunani~~~~uuan~odaaub~iiuu oz6un4iuau~n'uuanuuiidiuau~awir nidoqnhdnqa$ 4iuaudeani% iio n i ~ ~ ~ i ~ ~ i ~ u a z o o a~ a iC ~ idful u & o ~ ~ ~ ~ i ~ a ~ ~ n ' ~ ~ ~ l a n d'lu~i?%oC~moz ~ilu~iuau~awi~

dnw~uuj".~u~iuuo;l4iua~~awizriou~~~ , I.

15 r~u41uaud~zrrsu ~wrizd<ad.rznsuA~i]u~~an iio 3 uaz 5

aa (3

7s s a 1-

& s Q

5 -I

% llr

f f ss

o(a

--l'

,o a m

aa (

asf

"G=

... g *

4 -

9 (

jsa

C.I

aa aa o a

2 9

-a ca

5 C

a I

'"I

O(3

-C

2In C

(E .at E a

s =z " -6

PZ C C

-(G

." t d

4. ~71ai"uunI., 2. bra:

3. jiU4iuau

~ J P : ~ ? J L ~ w L & ~ti?< ~ -

44 0

asoli

Jm<

d~53d~:noudb~u~1uaubawi:

278

2. 4iwu~awi:d~ounii~~0~iin'u J7%E ka 2, 3,5, 7, ..., 277 u dpl

3. d~u?u~aw~:driouniiw"ao~iin'u J77077 aanudilioudiy 6a13miliuaurawi:G~dald

go 3 ua: 5 d ~ a d d i i3 ua:

iio 2 ui 2

MI^

77077 1Ja.sRi

5 WIS 77077 \Ja~&ai?bdu~%~

fiaiam14iuau~ir~i:GaQiaIddn fie 7 rdoaain 7 MIS 77077 n ~ g a i?4&.416 7

L B ~ ~ ~ ~ S L ~ ~ Y77077 ~ ~ W ~ J Y O J 77077 " t d 4. do1d6aimmi4iuau - = 11011. rlidiaun I., 2. uar 3. dn 7 11uau~awi:dliouniiwio~vi7~u ~ i i ~ im i iadsin u, n i ~ o u c l ~iio~ 2 , 3 ua: 5 a: t udn~mu47~aua:~iS 11011 1aiasGautiuoua~n"uu~~u~ 3 ~a~ku4ai~mi41uau~ar(i:Ga~aui

ks 7

bdos~in 7 WIT 11011 wsga ia~~fulG 7 ~iluGadl.r:nouBn~awd~vo~ 11011

drl

~iuau~am:driounii~~nrviiiiu JZB nanun~rioud~ako 2 , 3 uar 5 h:

udnJmu41uaua:991a

1573 lJasA~bGuou~ i n v ~ a o u4.d u ~ ~ ~ 1 1 m ~ f i i u a ~ ~ a w ~ : i a Q i a u i

iio 7 ud 7 nia 1573 Ybia~Ka i?~$u6o1am141uau~iw7:Kan'a!dBn iio 11 wuii 11

drm

Ro 2 , 3 , 5 uae 7 ~ i u a u ~ a w ~ r ~ r t o s n i ~ ~ ~ o ~ v iaanumliosdsln iiiu td L L ~ ~ ~ ~ ~ I U ~ W O ~ H 143 I T

~xias~aia~~uou-ain$uaau~ 5 flc7~~14iuau~awieda~muidn flo 11

l u 3 a ~ u i c ~ n l d . a " ~ r l ~ d a f ~ a u ~ a ~ a ~ f ~ a u l u n i ~ ~ u ~ri?cll3nis i 4 i u a u ~ ~ ~ iLucas z u u ~ ~-~ u Lehmer test b t m m ~ i ~ i ~ ~ ~ b Q ~ l ~ ~ ~ ~ f b % ~ ~2p~-~ 10 6~:1$41uauuuf~%u~a1wIj~u ~ ~ M ' i ~ 7 b b W ~ ~ ~ d

' L ~ ~ ~

8 ~ i b b & a ~ a ~ t i l 3 a a ~ o u i i ~ f l u 4 i u a u ~ ~bu~btilauhaifiu ~izw?a~~ W.H. 2546 ~<~u~~14iuau~~~izbbufL%ubb&a 40 (luau usz~~uaudddiuind~tilbviiAnnu 3o

6 &IUM~%

r.<>

2-11

G~~u~iuau~~.~izddwui~l~~uinn

& d i d

4.13

odni (36, 48), (17, 22) uae (-8, 0)

a'o&bna

81 a uar b b~udiuau~iiudJb~u~uu"Liwi8uiiu srl&ii b:

1. b~7mulab1mi (a, b) !&b~bluo uard~4 u ~ d i ~ a ~ ~ ~ ~ a ~ r i i u ~ 2. ( a , b ) = (a,-b) = (-a,b) = (-a,-b)

3. (a, O ) = (a, O ) = la1 ~ d aa 4. 8-1a l b ~r8a ( a , b ) = Jal 5. 81 c 1 a La: c 1 b

= (b,a)

uth c 1 ( a , b)

w a d h ~ ~ n d ~ r m m h ~ o ~ m m un.3.u. 1 3 ~aaa ~ ~ au uuar b o~lu;rlnlrnu~3a~Ku"uo~ a uar b 16 Ai;a (a, b) = ax

+ by

&~M?UYIJBI~~ML~~U x ~ba: y

&%&nm

1. 81 d = (a, b) laun a uar b ddluin

uh'hn13vn (a, b) a:ii!lin&

uonmnuunimd~uau~Gu x, y C(rliln' ( a , b) = ax + by h:

Aiid~mn7 fiotrli\$iuin~dun'u

mqrm

lnun1st.mu61 hmiu a, b

( a , b) ~~armidiuawdu x, y fhiln'

+ by L U ~ % & M ~ &a:ni~ai;dlunw~% ,,~~OJ~RQ~~ ~ d o ~ 30 ~i= n lO(6) + 5(-6) o r ~ i h i i30 ;r ( 1 0 , 5 ) , 30 z (10, -6), ( a , b) = ax

(6, 5) uar 30

81 m

= ax+bg

(6, -6) $ u $ a a r d w ~ ~

uRY3 m = ( a , b ) LTIL~UJ~~I~II 1641 ( a , b ) I m

l u n ~ w "(a,b) = 1

pinnqr*Jjun 4.6 mmuiralsu ..xu. ~ o sa ua:

b o!%u;lnmau

x bra: y b 16 $u& 1 = ax+ by hw?uu~sBiuau~$u z ua: y Gosnir~~amsi7(a,b) = 1 %uw7snKu% 6'1 1 = ax + by h~?uuis47~aaub~u u 2 d I ax asfa auuftii (a,b) = d o:!$ii d 1 a ua: d 1 b bauwqwlun 4.1 go (ii) ua: (iv) I$ $s$u d 1 (ax+ by) (I:!$ d dj ~ u d~ = a1 u ~ a s i(a, ~ b) = 1 qrhflunqwiun UR: d 1 by L%JL&"UOJ a LL:I

!$&t

(biaodig 4.16

~dosmn (5,12) = 1

(1)

asni

& o h 4 4.19 44 0

asni

15,17,28 ~ilud.~1uau~~w~:LW*ns'~Buf w u i ~ i l l ~ 7 i15,20, 7 31,43 ~ i l ~ d i u a ~ ~ a w i ~ & ~ n s ' ~ ~ u ~ w i o ! ~

~doapm (15,20) = 5

k$u16 15,20,31,43 !sj~~u47uau~awl:&w'nb'bflu$

. +, 3

l~mq~+ju 4.6 n INiG41uau~Ru x Lra: y $ s ~ i l r i '

bdosoln (15,17) = 1, (15,28) = 1 ua: (17,28) =

,.. r.

1 = 5x+ 12y ~dnssin 1 = 5(5) + 12(-2) luw"&$ x = 5 ua: y = -2 d Luosmn 1 = 28(12) + 5(-67) 6msnqwfjun 4.7 !$<--I (28,5) = (28,-67) = (12,5) = (12,-67) = 1

44 0

CLPd

bui~nisl~~duuuGq~7voa~~oa41uau~u naunn iri a bar b 19u4iualufudli~9u~uu"usr~Quuo~lu~d os~amimi6ani3iauuinvoa4iua~~f u1oa4iuau

a = p:p:

...p:

b =

LLA:

p,b'p:

...p:

bud pl, p2, ..., pn L ~ ~ ~ I U ~ ~ L O W I :pl& < p2 < ... < pn 11s:. ai, bi ~ 9 ~ P i ~ a u ~ f u d ~ i ~ i l u f i ~ i = 1, 2, ..., n 6aI.fulK

~doaain 600 = 23 . 31 . 52 . 11째 uar 2420 = 22 . 3O. 5l . 112 . 3=+.0) . 5m=(2.1). l l m u ( o 2) 6 I . f (600, 2420) = 2"~'(3.2) = 22 . 30 . 51 . 110 = 20

I hSY

4.t

~n a, w

URE

L U H ~ ' L ~ ~ ~ H( 1~4 ' I V IGUI IUR I ( 4 I

a 1 c UW:

LUH n r g r r r

runeuvuv a r ~ r n u

b1c

iidsl~da

u%r

2. 614~7uaubduu?nd aa a 1 d uaz b 1 d

dawmr.igmtau&ad~~tu~an"~~~a a uar b ri?u

[a, 4

luaj c

lurfiunds q il

m.7.u

1-11

lufiiusa~bua~uiiunia~116aniafauuir~

amimiRi~afau"iouvaa4iuau~Gu~oa4iuauh u

i ' n n i ~ l ~ ~ d u u u G ~ ~ 3 v o a w o ariibnauno ~ ~ uer a du'u ~ u W a us:: b 1tlu4~uau~Guuanuar~Quu~F]Ilu;d a = p,"'p" 2 ...p," bun pl, p2, ..., p. d

b =

UP:

~9u4iuau~awi:aap, < p, <

... <

p:p2

...p:

s,bi

pn uar

~9uPiu~u~Gu~Id~~us~

41 i = 1, 2 , ..., n 6aI.fulK -(s,b)

la, bl = PI-(q,4) d

Imun max(x, y)

...pn=(a.

~ ~ ~ ~s y rd

J, )

bnii (Carl Friedrich Gauss) ~n~~mni~mjc61a~uo~U'tb'b&w'aui b

~17%

do~nnun~ouddI ~ ~ % i ~ n q w j j l m7dni~nqaIj4iuauuu~aiunqaIj h6 &~iluddlmiodi~wnIuni3 Ji!ddrrqnd$

~uvi?%o~od\enhab~p1n~uix1 nqaljun u ~ m i i m a i u ~ ~ o ~ ~ ~ u ~ ~ ~ ~ o u f l d d ~ r

anainimoufi3~mof

$ii%tir"a ~fIu4iuau~it~aar m ~fIu4iuau~44~uan ~"iiu a mod m unu~~l~d!6~innism a &u m oin$uaoui'mimi.r a

ikuh amod m = r

V '

h u m ~rba~uoimudu q LLR; r dusaduak a = mq+r 0 5 r < m 4

~n.r~ni.rdinfjnui~11m~bbarnsiba8[iaaf~a~^a aoau.

62od194.23

3 m o d 5 = 3,

40mod9 = 4,

4 mod 5 = 1

bwm:

+1 9(-16) + 2

4 = 5(-1)

-142 mod 9 = 2 bwnz -142 =

~ h m w h uu P b

2004 mod 101 = 85

(mod m) w~9ianm

*hi.

w : o f b ( d n ) u n u a W n ~ ~ ~ ~ ~ & ~ b ~ n \ t n &?adid 4.24

ainunftaixl 4.8 ilzlgii 326 = 151 (mod 5) 6bd 24 $ 14 (mod 6) u~~~EJIG~I 326 = 151 + 5(36)

$o&islna

'Lw' m rflu41uaur6xluan a, b uar

1. a = b (mod m)

fidobdo

a m o d m = brnodm

2. a I b (mod m)

fidndo

!ui~4iuautRu k d~rh'Lw'a = mk

ido~qin 13 rwmrdi 3' '

c ~9ufiiu?u~4u qz\:r$iji

3 (mod 10)

huwqwlun4.9

(mod 10) $8

(v) 16

= 34 (mod 10) 16 134= 1 (mod 10)

~ a u m q w $ 4.9 ~ ~ ~Jo (v) 16 13' Tmunqwfjun 4.9

(iii)

(13~)'~'=1lo0(mod 10)

W$53'<, I*.?

13"

= 1 (mod 10) 16 1 3 - ~13 E 1 . 13 = 3 (mod 10)

%aunqwaun4.9 $'a (iv)

13401= 3 (mod 10)

- unasii dosmn

hunqwiun 4.9 &I (iv)

13'01

+ 5 = 3 -t 5 = 8

(mod 10)

~ u f ~ 13401+5 o MI~$~?u10 bw!or~[w 8

& a o $ ~3.26 94

5 = 5 (mod 10)

asw? 2644mod 645

arm? h s r n m i o

26U

(mod 645)

bulpinqdun 4.9

hunqwg'un 4.9 &I (v) la' (24)21 (16)~= 256 (mod 645)

r6osom

24 1 16 (mod 645)

a:1$

28 3 256 (mod 645)

rwmrii

(256)2 = 65536 = 391 (mod 645)

'Iaunqwiun 4.9 $'a (iii)

hunqwfpn 4.9

u'o (v) 15

(28)2= (256)2 (mod 645)

16 216 = 391 .(mod 645)

~obv9iua.s~~uai%!$(216)2= (391)~= 152881 = 16 (mod 645) (zS2)' = (16)2 E 256 &$%I$

(mod 645)

(391)~= 16 (mod 645)

rwnrauu (2a)2 = (256)2= 391 (mod 645) C

(22w)2= (16)~= 256 (mod 645)

umsii

(21")2

arl6ii

2512= 256 (mod 645)

2512- 21B = 256 391 (mod 645) nwrazuu 2@O= 100096 = 121 (mod 645)

~z!$ii &A;

- 24 = 121 . 24 = 1936 = 1 (mod 645) 2% mod 645 = 1

udlu+?&:n h a i

i~d3rqnmi35unun1s

ni3&sGa~~u~uuuurfiuu uw:ni~r$i~~Ga LI

nin~tnhadam~~:qn6fau$ustniiamaun,~~a~d~un~~latbni~~$ida$a;anianau~a~m ~o:wu!d sinn1afiiua~%a#l%rnm~n~"ar~wdmdasinni3~is 6114' k unu 68' srl& ~ k unura~dnriaunaiu41 ) ua: m rflu~uia~aamimaua~ lurnad$.P~ LYlXI:&

l6uhul%V%n'.a'~bbill%~&~ ~ I U , S ~ L L E ~kl:&Âśn'%~Wl% Y h f 6%uRd4dfkJ&ll#o h(k) = k mod m

lumrni~ardnhunaiu4i~l%~n"u$a~an'ufi~nini~~i'~1tbw~~ia~aam~'~1sr~a~~l%~iiu~a~ 4irfiuKaa '?'t

td'd

dauu%~flu~au4iuaur~wi:d~"~~ia'b~ss'a0nii 1.25 ~ l i i r a ~ h u a ~ $ a ~ a d n i ~ i ~ s t b f i a ~ u n ~ n u a n a u b w a n ~ n ~ d u

rd , ~~nin~~~%~a~dnri?unaiu4i~~uan'~~aanun~ian'un"%a~un'dn0ei~a~d~ii~iiwnimun'u~~0~iia 4

&aim 4.27

~uu~iis:uuds:uaawa'~aau~~nu.~ia'~~da ~ani.r~riu$agmuni~flia~~iaa~aai~~.~ww'na

Qiwu~ai~fiu 100 mu ~sluaanuuuni~~$id~~~nisu'a;aniafii~~udaa~aai~a~wCnai~~au

~6nisfliu~"11%aluuuuG~Iu 3~uu~waf ~ ~ ~ ~ ~ ~ w ~ ~ ~ l a i u # d ~ n a u & a u ~ a b a b56 3 9 1 ~ i u a t b n h ~OuiiGrhnLm~~$ida4qR fiQniumfiwufiffa ~~anuuusnuma~da~niiimauiiYuiw V

d a~~ i I % ~ d a ~a ~ i ~n f i ~~ : ~ id~ i 6 ~r l a ~~ I u ~i d a~n ~ u~ i i ~~u i

~~.~m~nauaad~~rn~au~aaddi~flu~iuau~~wi~Ad~uiw1,xiss'~ui 1.25 brii~oahuauu'o$a L ad

100 nun13 u u n n ~ ~ u i a v a a a i ~ ~ n127 a a 4arifltbdiuaurirwi: ~~ir~~ ua=fiwCnaiudaK~l

d3:hRj

"40123"

5i&asni3ii'u(iin$a;~~lnna~~u(iinGa~a~a~di~~nGa~a'bumimauig

A i m a u ~ m u i s a ~ i ~ & s i n n ~ sku &au n ~ 40123

uhhuaruni h(k) hmau

h(40123) = 40123 mod 127 = 118

Ituh ~ y a ~ o ~ w w ' n a i d d a ~ ~ ~40123 a : ~ i Gorgninrn'ufu~audnriaumaiu4isiii~b~da~ a 118 hbas

ludiuosr?! uan'u h(66893) = 66893 mod 127 = 91 h(50918) = 50918 mod 127 = 118

~uffa$aia~aaww'naiddaGid~:4i~a"66893" ua: "50918" ~:gn~arfiuIuunAnh

h o d i 4.28 ~

c a d

~ u n i s & ~ ~ i ~ a ~ ~ u ~ m u ~ a ~ d u ' n ~ r ~ . a " b n n ~ n ~ 6 a~unaiu?'iniluae uuu~~ao~wi~~o~~~ab~o

i h d ~ d s m h a i in'lonisl.a"nounpod ~ &flnua:~Buakrio!rlu 2' m ~ n i t n n d i u a u ~ 4 u u4i Biuau do m, a, c URZ q, l a u d m ~ i l J u q B k , ' Y a L J U ~ ~ cN~,1 I u d m ~ f i uuar ~ u 4 ~ f l u d i k u ( OLi: ~ 2 5 a < m, 2 5 c < m bbar 2 5 % < m u$a&~diiYu {%) ~ i n ~ u n i s 44

x ~ +=~ (ax* + c) mod m

&sr\6 {xn) ~fludi~~o~Biuau~~ud\xibfl~au laud o 5 < m nn 9 n w L ~ M IM rn = 9, a = 7, c = 4 bar q = 3 0rI6'Tdi A

+ 4) mod 9 (7q + 4) mod 9 ( 7 3 + 4) mod 9 (7x3 + 4) mod 9

+ 4) mod 9 ( 7 . 7 + 4) mod 9 ( 7 . 8 + 4) mod 9 (7 - 6 + 4) mod 9

q = (7%

= (7 3

= 25 mod 9

= 7

3 =

= 53 mod 9

= b

x4 =

q = ( 7 q + 4) mod9

+ 4) mod 9 (7% + 4) mod 9

= 46 mod 9

= 1

= ( 7 . 1 + 4 ) mod9 = 11 mod9

= 2

- + 4) mod 9

q = (7%

= (7 2

z, =

= (7.0

q = (7z,+4)mod9 q = (7q

+ 4) mod 9

= 60 mod 9 = 6

= 18 mod 9

= 0

4 mod 9

= 4

= ( 7 . 4 + 4 ) m o d 9 = 32mod9

= 5

= (7 5

+ 4) mod 9 + 4) mod 9

= 39 mod 9

= 3

~ d u g n wA do~$imkuhfio:~m~ou'bd~Bu$nwa D uarGagnwa 3 niqnniu l o X, Y usr r~ou'ld~iluGaa'nw7A, B

..., Z \dC~~gsl{ 0 , 1 , 2 , ..., 25)

a m i a r $ i ~ ~ ~ r u u Q ~ i f a : f i i ~ ~ u a G a ~ ~ ' ~ IaulG ~b~uG Aa ~ =n w 0, aB = 1,

~:~uiJji~lsluQoia~A~A.a"u f ~fluq~fi%uains'~~sl { 0 , 1 , 2 , ..., 25) flp) = ( p

94 0

asmi

C sliudi~u

UR:

Gao 4-194.29

= 25

s n

~oiu%nu

+ 3 ) mod 26

a4 ~ f i m ~ m ~ o n a"YESTERDAY" iu %aamwauuu(%%if uw

unuGai?nwafkli~ua6a u$a~s~'baa~u 24

uifa~$im~a~uuBmS ~~uLLnuuds:~iuau p

$4&~\6

~U~J~'% f(p) M =

+ 3 ) mod 26

1 7 2 1 2 2 7 2 0 6 3 1

sdo bmuna'~ ~ifltb~a$nwao:\~$onaiui~u'7a$~~sifa ~oz79'smm

"BHVWHUGDB"

fi~'7~9n~~6aauSauuuQgif IauliY4~~A.a'~ flp) = ( p 6auq4fiu

+ 3 ) mod 26

u5aiab~im~i~nnoaSa

f -'(p) = ( p - 3 ) mod 26

tula s~o:deu~a~"nwan~uui 3 diu~11.4~ ~~K3otii4

38ii

d' a ~~nu~ai?nwafffii~usl6au$3u~\aa;lu no 22 7 16 u w

ufjinoa%$auuu~~if ~auuwuu.daz4iuaup 'Lufidn'.a"u f-'(p) &4tua6

= ( p - 3 ) mod 26

~douwun$u~au~agnwa a:!$i$onaiu~ooaa~"auda

"TEN"

ainao~.io~i~ui~dua:~w"uii maL$i$auuu~~if~nai~dsoan'u'Uo4~o~su'ou ~ J

r n w k u i n i n d i ~ ~ ~ ~ u u uidu ~ 8 un~i ~n~ $ i ~ $ a u uRSA u flp) = ( a p

~$orn%$idis$abluuk ~ & i o i ' I u ~ d

+ b) mod 26

laad a , b ~3UFj'iwtb~d~ us: f duA~n".a'u~~ib~~v~d~do~d~ G~GailEii~

A = 0, B = 1,

gaaej19 4.31 91 .

...,

Z = 25 us: flp) = ( 7 p

+ 3 ) mod 26

b$imwmYon?iu "LOVEn ~aul.a"&d%u f

~~nuGa~nwa~fiipsusl6au$a~s~16 t o 11 14 21 4 Lnb$15Mok

flp) = ( 7 p + 3 ) mod 26 ~sluunu~1~auuds:~ p l %sa~;ln' ~a~ .a"u

&4tua6 7(11)

= 80 mod 26 = 2

~ U G Q ~ ~ &

+3 7(21) + 3 7(14) 7(4)

= 101 mod 26 = 23 = 150 mod 26 = 20 = 31 mod26 =

((CXUF"

mmuirn~i~uaoui'Nos~~G~'Ydwi (a,b) u a r 8 m i h u ~ u ~ ( uz uae y AllilrX (a,b) =

a4 o

asmi

u w

UX+

Luosqin

by lftaju

3553

+ 391 391(1) + 136 136(2) + 119 119(1) + 17

= 527(6)

527 = 391 = 136 = 119 = 17(7) ~j:ul8 (527,3553) = 17 do~dormbiuaut8u x uar y AllilrX (a,b) = ax + by

huniniiLun&

S ~ ~ ~ L A T I ~ I U I O J~ i ~ l n i ~ u n t 6 3 M " 1 ~ a ~ n ~ ~ " ~ o s(ASCII) a r u ~ ~ ~ o~Bufiu 16

Kaoeiisuatnq~j~nno:ndia

d o ~ d a : r f i ~ w ' ~ n m u ~ 3 n ~ d ~ u ~ 4 i u a u r ~ u ~ l a n ~ o ~ ~ ~ ~ a ~bohuada:rilu~a~iu ~u~ds~d~~u~n~u~ds!~ a:&srilu4iuau~~uuanduinnii 1 &nqajundoklw'

H~JIOLW~

nsSA b = 10

ir!Aii

+ .-. + %lo2+ %(lo) + a,,

a = anlOn ~,+~10"'

bun at E (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ~sor~!"iiuumua 6aiu &an-,... a,%% d

uarfiun a, 31 L ~ " u w S a ~ gs~ a,, I ~doba~~gn't~dau, a, ~BLWSIYI~G-IG~ ~a: a, Wa~a"~~iYn$oo rilufiu r$u

diuau 8963 dsulauzd 8 . lo3 + 9 . lo2 + 6 - 10 + 3

lun.rrii,da:nhai~ d

~ u a 1%

sa"UgipaaÂśJ

nsGd b = 2

AarrQou a l~uiiuosrG~la5u do

+ + q22 + q(2) + a,,

a = an2" ~ , - , 2 ~ - '

bodd~auniihai dusnos&a do

n i o 1 uarduurmu a Au ( ~ ~ h - ~ . . . $ % a , , ) ~

lu3zuun o u i r a ~ m i a z ~ ~ ' d m. ~. n u ~ ~ ~ a ~ ~ ~ ~ u b ~ ~ 9 ~tu~ i uua ~ z doo l~~ nr a~i uu ~ u ~ i u ~

& i wuuuduuunuCa ~ a a ~ u ~ z u u ~ ~ t ~@~ ni ui lA ' L w ~zmanlunirduudiu~uAddiuin 7 lu=@.aup~i%'!M"~uuu I. . e. de. du ~ z u u ~ a t 1 3 i u dw%3iuhwn a uaza~~~Up~i~~ici1u~~~araa4uo~iiu~ann~ar~t1 d

rdu (368),, (araa

3 azdiirBu 3 lo2, $qtau 6 azddirilu 6 10' uaz<a~au 8 $%ir?lu

Zu~zuu~at~~uao~n"irai~~ilra L+U~ i u (101)~ o . ~ r Cqb~q ~ u a ~1~ azJi7d1.4 1 . 22, Caraa

8 . lo0 azdiirBu

0 2' uaz&ratI 1 i d i ~ f l u1 . 2O b a m m u i m J i n q ~ ~ u j u4.12 n ui~~i~~diluudlua~~udo~~~3zu ~ar~uwiw!d~iuau~u3zuuraadnpuwit4

nounlmu.ddo$uld' ux

b (mod m)

laud m ~lludiu~utRuu?na, b ~Su41u?u~Ruua:z ~ S u & ~ l l a:don~oun?mudi%i s aounl~ou4 d' 1 3 4 ~ uatdun4iu?u~n"u 8~ s Cfaoan~~n'uux = b (mod m) 41 awaau ~osnoun~~ougmsLauu 6-P

ni.r~~~abaauao~noun~bou.a"b%~bdu m = b (mod m) r;il!6Malu% 6au 0, 1, 2, 3, ..., m - 1 f i m L B ~ ~ I ~ ~ M L ~ S J U ~ ~ ~ ~ ~ ? ~ P ~ S

asn7

a % ~ u ~ nnizbmudi o

J I ~

~1~w?wabaauas~nsun;bou6n"b%~~du 52 = 2 (mod 6)

h o $ w 4.38 P4 0

~ ~ u xd 6au i 0, 1, 2, 3, 4 bLar 5 5 . 0 f 2 (mod6), 5 . 1 f 2 (mod6), 5 . 2 f 2 (mod6),

8%!&$1

5 . 3 f 2 (mod6), 5 . 4 62

6 uabaauaemounfbou%w no

~ 9 9 ~ ~ 1~1i u~a~u b 9 &~ x

2 (mod6),

5 . 5 f 2 (mod6)

4 (mod 6)

E { ..., -14, -8, -2, 4, 10, 16, 22, 28, ...)

nn4iu~un"s:~iluua~~auao~noun~~~u4~%~~&~~i~un6au

$e&ana

% b i l ~ ~ ~ ~ ~ b a ~ " ~ 0 9 ~ o ~ n 2 b az 9~bdb b %(mod 9 b h m) b b t h * a*

q = %

+ 7cnz

I nuCt k ~Juiiu2u~bia.1 o:~~uwa~aauao~nown~~ou$b~9 ~auumau &ae$w 4.39

a~~iwabaw uaa~noun;bou6n"b%9bdu 2x

2 .0

otl6i1 &9;u6 l

$ 3 (mod 4), 2 . 2 $ 3 (mod 4) uat

3 (mod 4)

2 1 f 3 (mod4), 2 . 3 $ 3 (mod 4)

~ ~ u ~ ~ o u ~ ~ ~ G w ~ ~ ~ R u

& a a $ ~ 4.40 s

~~wiwasaauao~noun~boud~%9bdu 22 = 6 (mod 8)

?sP1"1 bbwudi

x &au 0, 1, 2, 3, 4, 5, 6 bbar 7

or!641

2 .3

u ~ n ~ i ix

3 (mod 8)

2 .7

bra:

6 (mod 8)

aat

6 (mod 8)

7 (mod 8)

~~uao~wa~aau~lrinoun~~oudiiuao~

noun.r~oud~3d~dii~i~um o i n ~ ~ m u & n d i r ~ i ~ 6 un~otu~n~~ u ~ oi iu $ ~ % n & r C f ~ i ~ u n o i ~ ~ r ~ ~ 9 ~ o lun.rd# ! i ~ w a ~ ama u ~ & rrsi~~iXin?+wbs bo F n u n i ~ i u a u ua 69r;iiIes"

t8u4iuau~6~~uan~id 9 1 ?&nnudin"+az!xjatnan ~~in aE

1 (mod m)

~ I ~ ~ ? M ? M LTi~ u M?d1!6

3z41 6 o ~ n - 1 ~7i ~ &l;i?%~i 1 7Si

b318rfiuniluaubbi~ Z u i l ~ n i ~ " t l ea9 aJaq1a m

1 (mod 4)

Fnurn.r~~wudiT i & a m 0 1, , 2 bbar 3

bn~~ni~AiriinuipliamT"~~arnsii,a~[iamf~aQP anau.

7 . 0 f 1 (mod4), 7 - 2 f 1 (mod 4)

&go

7 . 1 f 1 (mod4),

7-3

uar

1 (mod 4)

3 ~iluunifuvos 7 l w o q l a 4

d~fii%MY2E

lufiiuodb&13$u n'adnimi

bmuni3u~ud-1ii 6au 0, 1, 2 bar 3 2 . 0 f 1 (mod4), 2 2 f 1 (mod 4)

1 (mod 4)

2 1 f ' 1 (mod4), 2 - 3 f 1 (mod 4)

bsar

u a m ~ iIajiiwnKuvo~ i 2 IuuoqTa 4 hn%nr$i

a uar rn i d i u i n

ni~wiunw'u~muni~~~n~Fiiuu~t!aja~man ~ m m ~ i ~ a ~ i ~ u ~ ~

und$niimwd3:qnm

u a r m o r ~ i ~ n H i r u d d a ~ ~ u n i ~ b b f i n n ~ f l ~ b n u d ~ ~ d b ~ ~9in.ioriid 9o~n~iaia~d 4.41 ors~u-diwnifu~~~i~~ukadwidi!~~~~n!d floi3ai (7, 4) = 1 L L ~ (2, 4) t 1 flilG~31!6

?&!I

dodoin (11, 60) = 1 Tmwqwijun 4.13 164-1 wnifuvautloe 11 uoqla 60 n i r i i U nin~~mau~kod~n~m1~41

+5 5(2) + 1

60 = ll(5) 11 =

1 = 11-5(2)

= 11 - [60 - 11(5)](2) = ll(11)

oinaun13

ll(11)

buoaoin 60(-2)

+ 60(-2)

= 1

0 (mod 60)

or16-di ll(11) + 60(-2) = 1 (mod 60) &$u!6 ll(11) = 1 (mod 60)

u ~ m ~ i11 i b8wwnf%b~os11 uaqla 60

i h a e d ~4.43 ~

?8+1

~dwiwaraauao~flsun~boud~%d~& 1l x = 25 (mod 60)

9inGaOLiid 4.42 ?&4i 11 duwn&od

11 unqla 60

~ a n n u n l ~ o u d i r h n u a11 ~ haosdijl~ 11. 112

1212 d

buodnn 121

11 - 25 (mod 60)

275 (mod 60)

1 (mod 60) uar 275

35 (mod 60)

ana~ouriq s ~~6ii~~u~a~muvo~~1oun~~oudn"so'bai~auni3~~nudi x lunounpou&llinualdii

llx

6a$u1$ x

11 . 35 = 385 r 25 (mod 60)

35 (mod 60) ~~uwa~aauao~noun~~oudAfiinua

--,

gaae$7a 4.44

o~niwa~aauvoanoun~~oud~Qa~~u 3 s = 4 (mod 7)

oin$uaou?ko~~n^aa'b6ii

= 3(2)

= l(3)

+1 ?

1 = 7 - 3(2)

.iiifiounik!6-ii oinaunis

+ 7(1)

-2(3)

d n ~ l n n7(l)

= -2 = -8

3x -6x

Luor-ain -6%

-4

1 (mod 7)

(mod 7)

x (mod 7) ua: -8

asaoaoulaerni3untbd7 x 16 $u$o

&$il6 22(3)

0 (mod 7)

-2

ozlv;r"m

= 1

6 (mod7)

= 3-6

= 18

&$u x s 6 (mod7) 4 (mod 7)

x r 6 (mod 7) ~3uua~a~uao~noun~~oud~fiinua

~ m ~ r i a ~ i a a i n a i u sun q o - TSU ~ v i a i u f i ~ u i i~i 8 v o ~ o ~ ~ i u ? u n t ~ d ~ ~ 3i ~ onnao oni~n'3~ LRH

L ~ O ~ 1 ~ 6 - d5 i j~~Z L M ~ O L Q I W 3

L L A ~ L ~ O H I ~ ~ ' ~ U7

0:~~$066[kt2

b: ~ ~ ~ 1 ~ 1 ~ ~ 2 b 9 1 0 J ~ J ~ ~ "

finuiffmuiro~iiuua~ I ~ ~ ~ ~ : U U C I O U ~ ~ L O U ~ L B J L ~ I ~ L ~ ~

x x

= =

2 (mod 3) 3 (mod 5)

x m 2 (mod 7) w

o:~~uii~:uunoun~~o~d~~~~a~uuuoqla~3u~iuau~awi:~~~ndiiu~ilu~ rmo:niwa~aauan~a:uunoun3~auI d' h ~ l a n l i l ~ n q a ~ u n ~ ~ ~ w i ~ triouifu6oimmi~iodi~eioldu oaoatu

~nr~nirdi.ninoipliamf~~a::ndw~i~m~~aftf aeau.

&ad79

4.45

BJMIWALQIUVBJ~::YY~BU~~LDU~L%~L~

x x

384,

Iuda !&'I

(5,9) = i

18 z

+ 9z

= 5y

= =

3 (mod 5) 7 (mod 9)

laun y ua:: z r f l u B i u ~ u ~ ~ un u I u(1) I K

+ 9z = 5y + 9z = 5y dolain

5y r 0 (mod 5)

k&lK

3 (mod 5)

3 (mod 5) 7 (mod 9)

LR:

0 (mod 9)

Mar

7 (mod 9)

d o ~ n n(5, 9) = 1 launqadun 4.13 I6ii 9 ~iinw'u~uuoqla 5 uar 5 ihnwirluuoqb 9 ain&nou~o~~n~aI4 r i i~iluwnw'u~o~ i 9 uoqh 5 uae 2 ~iluunuirao~ 5 unqb 9

UUF~O9 lrs"

y = 7.2

1 (mod 5)

a::

5.2

1 (mod 9)

z = 3.4

LLR:

= (9-4)3

1.3

3 (mods)

kfit

5y = 5 ( 7 - 2 ) = ( 5 . 2 ) - 2

1.2

7 (m0d9)

~Iib?'lm'ji

x = 5y + 9 t = 5 . 7 - 2 + 9 - 3 - 4 = 178 aoanRuo~iiYstuunounz~eu.a"~Q~~h

G~~NIG 92 = 9 ( 3 - 4 )

(1) ~auni3n3aadimoua~u 178 178

= =

3 (mod 5) 7 (mod 9)

x x

= =

a, (mod ml)

a, (mod m)

1ma::~o15m?~awirn5ril,d(3, q) = 1 4 z ~ i l u h w u ~~~nuu I u(2) 16 V IY x = q y + m,z 1 ~ y~ LLI:: qy -t- m,z

= =

q (mod m,)

~daso~n q y

9 3+mz a, (mod mz) 0 (mod q) LLR::m,z = 0 (mod m,)

kiftbI6

q (mod q ) ua::

= ma =

a, (mod m,)

16orain (m,, m,) = 1 Iaonrl9.+aun4.13 I641 m, hnw"~'Luuoqlaml ka:: m, bq~niulua~oql~ m, IMYLilu y, L a MMRO m,yl 1 (mod 3) ~ a : : qy2 = 1 (mod %) IVY y = a,y2 ua:: z = q y ,

&&.416 q z

= % ( q & ) = (m2yl)q r 1 . q

lflu

q (mod m)

bbn: wy = ,(a,y,) = (m1yz)a, = 1 . a, = a, (mod m) (2) f l i l ~ Y 6 i i x = q y + m,z = q q y 2 + m , q y , L~uWaraaouoa~::uunoun~bou.a"~%s~~u do~dorua~aii5:uunounjboud~%a~h ( 2 ) daa~aasdualuuoqla m = mlm, a ~ a ~ i j idi ~ B u i n ~ a ~ ~ a o n ~(2) a u o a&$u!6 x' = q (mod q) x' = a, (mod n2.J L L ~ ~ J ? Ix' = x (mod q) &a:: x' = x (mod m,) q46i1 I ( d - X ) La:: % 1 (x' - X ) dnaoin (q,m,) = 1 n n u u u ~ n l i ~ u n4d 2.4 qm I (x'- z) L a uuno d = x (mod +%) '

Inrani~d~r?~nu~~riamfu~tnrii,a~[~aaf~~G'i aoau.

&adid 4.46

wrn w na~sdbfluuan6i~'au~a"uoa~ruu na~n~~audb~~~sv'u x

x 44

5 (mod 7 ) 2 (mod 9 )

?an? luC(ifLw' m, = 7 , % = 9 , o, = 5 LLR: 4 = 2

9rl@Yi1(q, q) = (7, 9 ) = 1 Lbar rn = wm, = 7 . 9 = 63 pl sn'o~n13~1yl LA: y2 n a a a n & a ~ ~ ~ n o ~ n l ~ 9yl a u= B 1 (mod 7 ) uar 7y2 = 1 (mod 9) 1 -3 ~Juwniuvas 9 huaq11 7 UR: 4 ~fluwniu"uo~ 7 Iua~aqI~ 9 L: 6'~uulw' yl = -3 baa: y, = 4 B Iaunqwaun4.14 !$ii x = 9 . 5 . ( - 3 ) + 7 . 2 - 4 = -79 b~uwaLaauwu3 u~t:

-79

<J~uI@Y

47 (mod 63)

47 (mod 63) ~~uwa~aaud~fluuan6ila"au~~vo~~r~unaun~~oudC(rfi~

" W

aio~isnms?9diaou!aa~u

h a h a 4.47

5 (mod 7 )

2 (mod 9 )

oaniua~aasC(~Juuan~ila"au~ava~3ru~lnaun~~ou(n"~Q~~~ x = 2 (mod 3) x = 3 (mod 5 ) x

2 (mod 7 )

m -

uat M3 =

m, G a ~ n i a ~yl, i y2

= U A

105 -

= 15 7 d ~y3 naamn$aafunaun~~au(n'

35y, z 1 (mod 3) 21y2

1 (mod 5 )

15y3 z 1 (mod 7) at16-h

yl = 2, y2 = 1 unt ys = 1

la~nqw#unc~rw a"uaa"ap1 ~ ~ ~ t lG~a~aaer4a

x = Mlqyl + M2qy2+ M3a,y3 = 35 - 2 . 2 un:: 233 = 23 (mod 105)

&$ul&

+ 21 . 3 . 1 + 15 . 2 . 1

= 233

23 (mod 105) cIluu~m~udciluu~nd~w"au~a"~~aazuunaun~~~u(n'd~~~um vw

mui3nm.raoAimoulmaau 23

2 (mod 3)

= =

3 (mod 5)

2 (mod 7)

(a mod q, a mod n,..., a mod q)

&?a$?s 4.49

fi' m,

rilu47uau~owiz~uw'mf~Bu~ d lslunqw~u~wnloaa~iumui3nrr~n~!&i~ udazhuau~lu a 4 4 o < a < m b u m u d a m = 99 98 97 - 95 mui~numu!$i&u n - id~u~1um~dub~rwa~io'~1innimi3&au mi !&uuu~iuarv-ii& rJu = 99, m, = 98, m, = 97

uw m,

= 95

a = 123,684 mui~nduuumu$iau 4 - $ ~ o Y do u~~J

(123684 mod 99, 123684 mod 98, 123684 mod 97, 123684 mod 95)

&I&!&

a = (33, 8, 9, 89)

~ufliuodb~ua~umui3nunu b = 413,456 6au 4 - iho'ra6~do (413456 mod 99,413456 mod 98,413456 mod 97,413456 mod 95)

b = (32, 92, 42, 16)

~i6osnimi a (33, 8, 9, 89)

rs?m~i~nn~!&.nuni~b% 4-

G J ~

+ (32, 92, 42, 16)

= (65 mod 99, 100 mod 98, 51 mod 97, 105 mod 95) = (65, 2, 51, 10)

$ i i ~ o s n i s ~ i i ~ 4 7 u a u ~ a # ~ w u4&-a&~u6uif u bmn7!d~1uni3ni wabaau.ao~ 3zu~naunfbou6

x x x x

= = =

65 (mod 99) 2 (mod 98) 51 (mod 97) 10 (mod 95) cd' a

~ ~ ~ % ~ ~ ~ ~ ~ J M ~ ~ M ~ ~ Y ~ ~ ~ M ! $ ~ W ~no ~ ~537,140 ~ U Y O ~ ~ L L I ' U ~ B M ~ ~

ddsz!~wa~w#b~urduan"un"u 123,684

+ 413,456

= 537,140

oh^ 4.50

d o w n 101 ~Ju4iuau~awi:uar ( 2 , 1 0 1 ) = 1 oinnqw$unuoaud%'uifl& 2''"

1 (mod 101)

ni%bGi%%%bkuu RSA *

n. L J , ~,~quunaau

3 i i u l o Ronald Rivest,, Adi Shan

ddia~in 6aun~~udaaudatCao'nw~Uu6ara~riou~~dauiiuni~~~i~$~uuuQms" rrAiu.i~n(uAwun'?unais~ uiaaiud&oani3 kda:n~uo~~i~auarunun(uuoaGaCnw3uiaia~%13$~bun13&~fiJl6%uni~udaaa$1

silu d

6auw p bar q ~ ~ u ~ i u a u a a w i r1s a<~ e < ( p - l ) ( q - 1 ) ua: ( e , ( p - l ) ( q - 1 ) ) = Caiaoeiia

&?adid 4.51

~~udas$anaiu "STOP" Imu1I3Ka RSA d o p = 43 uaz q = 59 e = 13 ua:lts' A = 00,B = 01,C = 02, ... , Z = 25

son

-4

35911

Q:L~U~I (e, (p - 1)(q - 1)) = (13,42. 58) = 1 ~lin$anaiu

STOP uLflaaElraGa~a~14'

18, 19, 14,15 ui=hskn?uni~a~~ilu 4 ngn (~uo~~u$aanaacnii~$i~$~;5~) 14'~Elu 1819 1415 uda~s.n'pibsldI C = M3mod 43 . 59 ~ $ 8 C = M3mod 2537

6JI.trali 181913mod 2537 = 2081

ua: 141513mod 2537 = 2182 &I.tra<a ~ ~ ~ A l 6 ~ i n n i ~ ~ ~ i s ~ i a ~ d r a 2081 2182 ~ $ 0 uga udaa~duGa~nw3!d~Elu "UDVE"

-*-

nbA

I I3ZIUW3YIQUUU

~nmuianosnaC~s~aiud~.~'i~~ RSA ~n~na#vtTink&s~uonnl d

~uoJoin(e, (p - l)(q - I)) = 1 &$u~i, de

a:ddiu?u~iiu k

de = 1 @

~~LOUW~CU e Yluuoq ~J

1 (mod (p - l)(q

+ k(p - l)(q - 1) = J@ =

(M)d

a (p - l)(q - I) 16 uu

- 1))

&~$ul~

M+Mp-l ) ( q -

(mod n)

Its' (M, p) = (M, q) = 1 oinnrlwiunuos~~r15~16;ja641 W 1= 1 (mod q) MP1 = 1 (mod p) ua: " & ~ ~ u Cd = M . (Mp-l)Mqm1)= M a1 E M (modp) @

bb8:

M . (Mq-')MP-')

M. 1

M (mod q)

d o ~ o i n(p, q) = 1 n n n q ~ ~ ~ w ~ ~ ' l n a n ~ $ u 1 & 4 1

@ &U;â&#x201A;ŹI

L37&~JMl~i P

M (modpq)

P = @modpq

6 a s O i ~ d ~ ~ d o : u ~ n ~ n i m RSA saa~~ &ofh9 4.52

0981

0461

d w w

r~$aoJno@avk RSA fiaanym~M"1usnd7 e = 13 ua: 44 0

asni

oin n = 2537

uun6ada:nour3diu?urcbwiz 2 &aagmt7u!K

mo:~ih d drauwniumuan~ e = 13 l w o ? f i a 42 - 58 13d

1 (mod 42 . 58)

~ 9 s 13d

n = 2537

n = 43 . 5 9

&uiioni~id d~dilts'

1 (mod 2436)

I n u n i ~ u h u n p o u 8 b ~ ~ ~4.42 u ~ ? so~~i1~6d = 937 O O ~ T ~~~U&IJMI~I ~ H P hilts' P = C'mod pq IWJUVU~I ~J$MI~

C = 0918

w?o

P = C937mod 2537

= 0461

O98lg3' mod 2537 = 0704

$saaiu~kuiiis 0704

1115

uwz 0461937mod 2537 = 1115

~~non~?lu$anaiu!6rh"HELP"

~~%735~7Td7 13 M I T ~ I U ~ M ~ ~ ! ~ J ~ ~ J ~ ~ M ? O ! ~

1.1 2.

1.2

197

1.3

247

hw?u41uau~lu a, b La:: c tsl q osfip6ii 2.1 61 a ;t 0 &;a a 1 0 LLR: a l a 2.2

a 1 1 fida~idla a = 1

2.3 2.5

61 a 1 61 a 1 61 a 1

2.6

87 d = ( a , b)

2.4

a = -1

un:: b 1 a ua'a a = b a = -b c rm:: b 1 c h u f ! ( a , b) = 1 uh ab 1 bc un:: ( a , b) = 1 uga a 1 c b

bsga (-a -)b d' d

= 1

3. osfioiamii1~1~~~6~!d$b~u~1uau~a~i::w"sab~u~1uaud~tnsu 3.1

3.2

3.3

101

4. os~oiamii14iuauds!d~~ilu41uau~a~1::w?s~ilu4i~aud~tnsu 4.1

4.2

4.3

1011

5. o s s s i ~ n s s i s ~ n : : L ~ ' 1 3 ~ w ~ s " ~ s s ~ i u a u $ ~ 7 w u ~ 5.1 5.3

w i ~ 6 a u 12 78 ss156au -13

wir&au 4

5.2

-1

5.4

-999 ssia6au -23

6. oswiwsn~auwr~ar~~woues4i~au~Rissusl 6.1

2 wi36au 7

6.2

-333 wi36au 33

6.4 2345678 wia6au 101 ~ 1 3 6 a u-89 7. oswi~adsznsu~~~u41ua~~a~i::"~ss4iuauds~d$ 6.3

-7007

7.1

7.2

729

7.3

2222

7.4

505050

3333

8.4

808080

8. osssi~adel.r::nauff~ilu4ipaau~awi::~ss41~aaudo~du 8.1

8.2

987

8.3

9. ~swi4iuau~lu~lan~ra'aunii 24 un::~iIIuBiuatb~awitCuw"nd5u 24

as~i~iuau~GuuanA~"sunii 35 ~~nt~~u~iuau~a~i::IuW'nd~u 35 11. os~oi~m1iihuadlhwua~u~~6n::~sds!dd~~u4iuaub~~i::~uw"nd~ub~u~~?s~i 1.0.

11.1

17,35, 64

11.2

28,33,79

11.3

19, 29, 39, 49

11.4

35,46,57,681

12. a~~nrmi~14iuadrhnua~u~~da:~oda'Ldd~~u~iuaurawi:~uw"Y1f~u~~u~n"~o~~

13.

12.1

3, 5, 11

12.2

38, 43, 99

12.3

22, 47, 89, 101

12.4

35, 46, 57, 681

~ ~ . L B R J3~ I ~ 1 1 ~ a ~ ~ ~ 1 1 a ~ ~ 1 u a u L i ~ m ~ ~ i u a ~ d d s u ~ ~ a ~ s a ~ ~ a ~ & a

14. M m ~ i l d ~ w u ~ i u u a n~ u ~ a m a i nirua~am~4iv,a(?ru& ~ d

15. ~ i u a u ~ u 1t ~CJ~ 200 i d ~ 9 ~ ~ 6 1 q~ ~4~ u1 o~n i7r 6~a 6u

m 41uauC(dsut~anii[~~~~iu~6a~~a

~~FJLHGBLP~W 2

ua:~dovr~sKau 14 uga

L M ~ ~ O L10 PI~ d d % J I U O W I ~ ~ ' ~ U7

16. 4 i u a u ~ u t h ~1~ 8i 9 300 !&n~ad41uau

u g a ~ n ~ a ~2~ ~[ BwZ L ~ O Y \ I = ~ ~11 ~ ~ uga ?U

L M # ~ L P 8~ W 17. 1% m ~ V u i ~ u a u ~ i u u La: a n a, b $ti a

LA:: c L ~ U ~ I M ~ M La~~ U filadji

c bc~ (mod m) uga a a b (mod -)

(c, m)

21. a~ni&ni~+aaJil1nua:&a~mfa~]~0u1189~iua~~~ilbbd~~:~eia~d~ 23 . 32 . 5', 27 . 33 . 52 21.1 21.2 33. 7 5 . 1 1 2 , 27. 3 4 -72 21.3

413 43" 537, 413 . 432 - 53'

21.4

3 - 101, 0

22. a~n1di4iuauda$dif 22.1

23 mod 3

22.2

-99 mod 13

22.3

2340mod 341

23.2

-97mod11

23.3

3340mod 341

23. a~n1d147uaudo!df 23.1 24.

13 mod 3

~ J V I L ~ Y I ~ ~ ~ M ~ 2350 $ I ~ Y ~ J

25. o~ n i r a u ~ a a ~ a s n ~ n w '3*O iu~o~ 26. a ~ q g o d $ ~ a = b (mod m) naorus a mod m = b mod m a ,

27. ~ ~ ~ 7 d 7 U M i ~ ~ u n i 3 u 4 1 a l u ~ 7 ~ m u ~ . a " ~h(k) ~ n '= . a 'k ~mod ~ ~ " 101 11

% ~ ~ i 3 ~ ~ L f i ~ ~ ~ 4 1 ~ 1 0 f s l Y ~ ~ ~

i~f~nualfidoldd 27.1

104567890 (65)

27.2

234056793 (100)

28. -a~~id1r~nCb~~un~au~laiu41'1a~~.a"~~~.a'uuero6 h(k) = k mod 101 lun7rcmrfiu~rnaafm~a;lCn#nw ~ R I M U ~a ~ l dRd ~

28.1

610235872

28.2

330126981

29. o ~ n ~ d i l ~ ~ ~ a ~ ~ n n i r a ~ " ~ ~ & a ~ a ~ ~ ~ ~ ~ u u ~ ~ u ~ ' I a ~ d & a r i a f i i ~ ~ ~ ~ l ~ a xn+, = (5xn

+ 1) mod 7

LUO~ILP%U~~ q

= 2

lassmsiim~nui~1iaa$ua:n3aariaa~~aG% aoau.

30. o ~ w i r l i b d ~ ~ a o i n m a & ~ ~ a ~ a " ~ ~ u ~ ~ u u ~ ~ u u b a u d ~ a r i o i i i ~ ~ a d ~ d o i n n o u n ~ ~ o u b a ' ~ 8 ~ ~ ~ u d

~uociiku6'uq, = 3

xn+, = 4xnmod 11 31. od&T$&naiu

&eidLld

"SEE YOU NEXT TIME7' hul%Ad%u f

31.1

32.1

f(p) = ( p + 1 7 ) m o d 2 6

31.2 f(p) = (5p f(p) = (p 3) mod 26 32. ad~fia%%onaiu UTOMORROWn laul%A~fi%uf &dd(d6 32.2

f(p) = (7p

+ 7) mod 26

+ 3) mod 26

33. odnoa?$rdonaiu "JR WR WKH SDUWBv ddgndauil~unis~$if~'ab~u~l#%i$ "QHAW ZHHN" d~gnti~ui~aunis~~h%M"auuu~%7a"

34. adnoaT$&naiu

35. a ~ ' L + ~ u a o u ~ ~ o ~ ~ n t ~ w 1x4 ua: i u a 3u ~AYI~IIFu

36.

(a, b) = ax

~uo

35.1

a = 18, b = 12

35.2

a = 201, b = 111

35.3

a = 1331, b = 1001

35.4

a = 9889, b = 6060

35.5

a = 53421, b = 12345

35.6

a = 11111, b = 111111

~ ~ d d ~ ~ ~ l ~ ~ ~ d f l M ~ ~ ~ ~ " b ~ ~ ~ " U ~ 7 ~ ~ b ~ ~ ~ 7 ~ 3 ~ '

36.1

123

36.2

4532

36.3

100632

37. ~ndduudiuau~rhwua'L~"buba"~~i~baod~~u~iuaoa'L~b~n~~~u~ 37.1

37.2

1 111

38. o~dduu4iuaudrliw~ 38.1

103F

38.2

1 0101 0101 ~

37.3 ~

ABBAB

111 1100 0001 1111

38.3

BADFACED

39. odbdduu4iuau~finua~w"buba.u jiuaod b8~6iuau~u~a11~7u~uwn 39.1

1111 0111

39.2

1010 1010 1010

39.3

111 0111 0111 0111

40. o ~ ~ ~ i l u u ~ i u a d ~ ~ u a I a Y u ~ n " ~ ~ i u u ~ a ~ ~ u ~ i u a u Z u ~ a " ~ ~ i u ~ o ~ 40.1

7345321

40.2

12345670

40.3

100063211

41. o d ~ d d ~ u ~ i u a u ~ d i a u a ~ ~ u ~ a " ~ ~ i ~ a o d ~ ~ u 4 1 u a u ~ u ~ n ~ ~ i u b ~ d a 41.1

1011 1011

41.2

10 1011 1011

41.3

111 1111 1111

42. oduaadii 15 ~8uun$u"~a~ 7 'Luuoqla 26 937 L8uwn$u"Uod 13 huoqla 2436 43. odb~aa~ii 44. odwlwn~u~~od 4 luuoqla 9

45. odniwnwi"~od2 luuaqh 17 46. od~i~nW"u"~od 19 luuoqla 141

h 47. odmwnWiruod 144 ~ u u o ~233 48. a ~ n w a ~ o a u u o ~ n o u n ~ ~ o u d ~ ~ ~ ~ ~ u e i o l d d

49.

48.1

48.3

lox

5 (mod 9)

7 (mod 13)

a un: b ~ S u ~ i w u ~ T G u,m,

I$ LR:

..., mn

m = q q . . . m n odkua~ii 61 a b (mod m)

7 (mod 17)

48.2

48.4

6 (mod 11)

Lilu+iIi~?ub6uuan%~(m,, m,) = 1

b (modmi)

L ~ Oi ;e

i = 1 , 2 , ..., n uii?ozl6ii

50. oaMiwa~aau~anumYo~fzuufloun~bou~~8~~~ud~~dd 50.1

50.3

x x x

2 (mod3)

4 (mod 5)

5 (mod 11)

= = = =

1 (mod2)

x x x 50.5*

x x x

= = =

50.2

50.4

= = =

2 (mod3)

= =

5 (mod 3)

8 (mod 5) 4 (mod 7)

1 (mod 4)

3 (mod 5)

2 (mod 3)

3 (mod 5)

4 (mod 11)

= =

7 (mod9)

3 (mod 10)

4 (mod 12)

8 (mod 15)

= =

5 (mod 6)

50.6

2 (mod 4)

16 (mod 21)

51. ~a~i4iu?u~6uin?s&'au 3, 4 uaz 5 usa~n"Rormt2, 1 uaz 2 a 7 u i i h 52. oa~i4iuatb~~~Ani~$iau 5 saga ud~~oni't6-~u 3 LL~Y~L~~~1 LPIH 53. wu~madi 2340= 1 (mod 11) 54. flauamadi

2340 = 1 (mod 31)

55. sauamaii

2340 = 1 (mod 341)

~mu~$wqw~unueaudfui6[uu&

2340 = (210)34]

1slubi?nq~3uw"uoaudi~i6[ U M L ~ I2340= (25)68]

56. w ~ ~ ~ q ~ ~ u n " u o a u d ~ u ~ 6 u a ~ r n q ~ do ~ nuinh ~ ~ 1 ~ 3 ~ ~ ~ o o a u 56.1

3302mod 5, 3302mod 7, 3302mod 11 uaz 3302mod 385

56.2

52003mod 7, 52003mod 11, 52003 mod 13 uaz 52003mod 1001

57. sani a da~9u41uau~6ui!d~iluau u n ~ ~ o u n28 i i buduuodlupJ (a mod 4, a mod 7) 57.1

(0,O)

57.2

(1, 0)

57.3

(1, 1)

57.4

(2, 1)

57.5

(2,2)

57.6

(0, 3)

57.7

(2, 0)

57.8

(3, 5)

57.9

(3, 6)

RSA ~oflaiu "ATTACKn laud n = 43 59, e = 13 59. aanomnki RSA ~da&naiuAIm~~~ 3 o 0667 1947 0671 Tau# n = 43 -59, e = 13 58.

P~L%IT%

44 0

aan7

wuuGaia 3

k o d i a 5.2

BIM fin "111,

a2, a3

uu C

~n~aumin~i%idda"d~\?'61uunysundnua 20 h ~ i u ? u ~ ~ a i u n u n & % n ~ u n i d ~ a u ~ ~ sinYjiPiidd6aiauunyi un:~~umanrillain~~~3'wuuny"au~~iiPii~bwud~in~~~~~a 4den

nun:di~u~ildu n a ~

94 0

?an7

kbaa~un~n~mud~an*b~u~u~%nao~b'p~a {O, 1, 2,3, 4, 5, 6, 7, 8, 9)

aani

m ~ d n ~ ~ o ~ n u o z ~ d d a : ~ u ~ s i i ~2$ ~i'a ~ ?do' a ~ ~ j ~ ~ ~ ~ d lm 1 n2wo~nudb~wm~n~3u~~u~ddazvu'bd' 1 G d am 2 n Z ~ o ~ n u ~ ~ ~ ~ u m d n n u i C u I d d a6z V%u l d ' u h u a u 3 ~ i ~ n j [ d d ~ : y d i 1 ~ ~ ~bvi11-7~ n a " a1 + 6 = 7 53

uiuadidrdou 4 nu ~ s ~ v i o ~ o ~ : ~ q l ~ ~ o u u i n i u o i ~ i ~ ~ a u n " u ~ d s ~ u ~ ~ o ~ 1 ~ i

h a dm 5.8

s~dt'a~%~dou$s~~miC?'a a6

2ani

i~i 1 mfilyr60urndsnu ii -4

am 2 a~r%~rdouaosnu d 6 55 d am 3 r~ir3ql~dousnu~u ii 4 i'a

PII

a m 4 raidryrdouinu8

~iuauamsnumduiuaiio~rt~rdoubvii~~ 4 +6 ra

+4 + 1 =

i'a

uisnasra76osni~n.nu4iu? I~niditTnw33 Ka60 A, ABC

ACB

BAC

CAB -.

CBA

nisfi'm~uss.uisiu~~buaii~~sd

~ i r ~ ~ ( i sac~iluo'nw d1 A, B A s C fl6d3KmIk 3 t 5 ~ I U M ~2S b~#a;n~~i~t1%70~1fi'mds~~~flu~ ~ 2 ?fa4sfi'aIii' 2 ":a di~r~(isd 3 r~#otTn~,arius 1 6ai7dsfi'alK 1 33 63th

aII

i i w u a m s ~ ~ m ~ v3i lx~2u x 1 = 6 ?'a

nir~mfios~~KervosGa~nw~ rdude~uni~ ~~~"aersi~~~~~dplw

~ n s a n i ~ d i n ? n u i ~ i n ~ i ~ ~ a t ~ f i a ~neau. [iwaf~~ii,^n

RIWU~Gn ~fluBiuau~4a~uan U ~ R P ? O L ? L In~ ~;ijuu~~nu&au n! 'baud

n! (n- r)!

= -

k o d i s 5.9

d i g a 6 n w a i n A i i 7 STAND

ui$@~~a~~u~i!nBi'bslu~Bi4i~a~~aaiu~uiu ~&$ana~a~ii

h u n b d r f i i h i a a o a U n - ( r - 1) = n - r + 1 i?! a

' ~ I U ? U % ~ $ M ~ L I I O rJ

bdlcu

n(n - 1) ..- ( n - r

+ 1) =

n! -a an ( n- r)!

n'7d&va~ n $~uandi~iiu$qnua~41uaui%mfiusaurd$uur%sr&rviin'~n! 3; ~ u u 3 n h on~

lfJu n, brndouiiu~iluniud1

L r n i o u n i n ~ ~ u n ~ku db u d n, + 3 + ... + nk = n

d n, ~ ~ ~ d o u ~ u r ~ 2u n...j u uaed d nk II~?~~'~~s~~~L~~~uuoEM'DURJ

n! ihnaui~sdritldiuudl~uand~sCu q! 35 hnit6~vo~rndounivosn~ud 1 ua: d

&vodmdouniruo~niud2 ... uac ni i f ~ . m ? u L o ~ r n ~ o u C u v o ~ kn ~ u n

GiJuu ~"a~s~u~diuudrnu1anrdunaiuuand~~l6ri1iiu

ncl

aa rnnQ

106

aoau.

P P I PI

a n ~ . r u s ~ ~ ~ ~ i u u ~ ~ ~ a s n a u o r ~ i i ~ o i 8st~wmr5iifiusfi~~1t n ~ + i j ~ ~ f i ~ ~ iuu~q~a~nau!ii~i? ~u~~s~~ 3! = 6 53 unarriahouna ~ d uauuG.ilok6-11~3 h i i o A, B uat C fius19ud~~aiozfi'a!G8an~~m ;â&#x201A;ŹI

ABC &B

BAC BAC

CAB CBX

uifi'mhw~ia3 si?fius~~uasnauo::d?+<a~~u~ 2 i'nriik &$

orbt4*u.din13cmfiua ABC, BCA ua:: CAB ?obgu?"a6!uan'pbuarni5cmfius ACB, BAC uar CBA fin~flui?iduan'uk $ u filhnumlG A o@aiormlohwtn 2 PPl w

ai l o B uar

amm16 2! nio (3 - 1 ) 5?i oiiYuwafim~~luZuaiYu~ mu-irnqd~G~sBi$ldt

driosfiusfi~d~uu$sd -

~4 "saiinir~anu&~s~uT~z n5ornr$a&?

' ? ~ H ~ L ~ ~ ~ J ~ ~ ~ L ~ ~ ~ I ~ ~ ~ ~ ~ J BuaL ~ud~ O ~ A X I O J

n ' ~ n i k d u ~ h i l l d ~ u u ~ ~ " ~ a ~ ~ ~ u ? a n a u n " ~ ~ i~du r ~ni6auwa~ui& w ~ n n ~ ~ ~ ~ nnir?aiot;nfian'% &iu~d~~& w?Jqyuo Lau%u 6rai~aini3d3u~Kaj?nw~ 3 Ka A, B uaz C ~3u?~naun"7ua~Au~ioarazdni~CaSu~Y[$i 2 uuu

~ n r d n i ~ d i n 3 n u i ~ i a w f u 1 z n f i m ~aoau. i~61~~~~f

fiwtn8n&ii7uvl~dni~mdaudABC azma~r%ni3<mfiud ACB

K~$%~IPC?UP~%I~&PSUJO~T~W~LBW

~4u~uuu~#ua~riiUi7u

3! =

i%~-ai~nrtni=ibiua~a$nw'5 4 6a A, B, C 6 33 &id

&RE 4,

D ~il~?sn~~dd~60~0~6iub~uaazdni'i<a~ud~6

-* ...+

auiQn 2 6a a s ~ ~ a a d d a u i3~ n 6a P

auumiuLam A = { a , b, c }

,- a % & ~ a a a o ~A ddauian 2 maunJnun 3 h ~ a lao { a , b } ,

riiwo~~0iuanirfunisrijon6aunu$nfiuu2 nuoin~aGn.r3 nusrd'it~iion~~nun 3 5% aa~ih1amaaoanraon6a~nuu'nfiuuG~nEjia fiunii aaomplci k m a i i b ~ ~ s l d d a u i %2n Riao~wn A u"i~diu ~M~YwJ&K~MT~u<Y ~~n'bxiGnaiuh~yrwm:ii c , d t ~ ~ s l a o ~ ~ ~ a ~ v i i n w n a o ~ u ~ d a u im % n~ ~unura~aupan~n~d s : ~ ~ d i u ~ n ~ o ~ m u a n ~ ~ ~ a o ~ " ~ w d n i a f i u ~ h ~ { a , c } uar { b , c }

all

npl

r a w

r 4 8 - w

o:iiwn~auGuao~$~ao~ diuau~bflu~&udduuo'n~~ 2 ~isin$na~$~num 3 diuau7Enn~Knaa2 ni nnKnw~$~nun 3 Ri d

Pn,,= (n - r ) !

3 fl

P,,, =

6 3+

Cn,r

= -Pn.7 =

r ! ( n - r)!

iEvh 6i~un%%q 2 nu AGe~uiri?use~%g~Aeunwdu1~~A~~ 4 nuain 8 nu

udii7wn%i~%qldh~ 2 nutuui~auarsio~~%ry~Aeu 6 nuain 8 nu

( )

huaui"a~%g~deu~miR"u

PCI 0

aam

= 28

diuauihlugnuaakm 1 gn uar'iiu 2 gn ~riin'u

8 (2)

= 112

iii

{a, b)

uaz {c)

{a,

uar {b)

{b, c)

uar {a)

i uuaRalurn&lPiuauit~a~$an6agna3 2 si? ~Jun(uu~nriauarl6~16ao'n~3~~~1aoarrJta6a~n~~lunrjud 6

uar n,+n, = n

biuaui"s~i~n(u~vi~n'~ w

n! (n-n,)!

- n!

q!q!

aiu~~~osumneiimi% n iJrio.sni3ud~~~u 3 niu 7 a: q uar % miudi& bun n, n

~ i u a u ~ ~ u e i n r j( u n li

gaedw 5.21

a r u i ~ n u10 aiud16iJ

)(

n-n, rr)

)

9% b"USIgn~uh~ 3

n! ( n - ) !

-TI!

(n q!(n-q-q)!

~ m ~ d u e i a f f 2l onu, 3 nu uar 5 nu

I ~ - d ~ n ~ ~ d i ~ ~ n a i m ~ ~ i ~ ~ i aeau. rnrii,~~i~m

h a i i 5.20 ~

o~tndiuau?+$anu 6 nu diufi2lu 2 una q i r 3 nu bsdnuie~uuanuiw~nlo uirndo~lmanaj8ur3u~una

( a + b)3 = ( a + b ) ( a + b ) ( a + b) = ( a b)(a2 2ab $) = a3 + 3$b+ 3ab2 + b3

n rilu-tiiu?uii~iiuin7 Q:G~I~Tu?I~~I~ (a wadi 1

wadi 2

Qhnm

+ b)"

= (a

+ b)(a + b)(a + b) .;. ( a + b)

Vknnninti a ainyna~clu+jmn"uln'an d

I 6 a i n n i d i b ~inndrm(lu+jwn"ua ain ( n - 1) wriiumwao

1. Piuauwal(ainnims:alu 2. lummaaiu ( a

( a + bin dAnua n + 1 wad

r a ~ B f i i i % ~ao Auain ~ n u$aama.sia: 1 o u 6 ~o

114

Ins~n13d1n'ineriniaa$~rarnSii,amaaf;aC3aoau.

iaura.lr&ii&mos b duoin o ~4uGuiar 1 ou6s n Y 3. %ubdat~9d.lr~~n13n3rol~ (a + b)n w ~ ~ x I . ~ ~ o J L ~ . ~ ~a . ~Uat ~ ~ bI ~ Lnlfk J " u on ~ Lauo 4. wodd r

s i ? o d i ~5.21

mnsrnm3roiu (a + b)" Go

oslnwodd 4 .lrosnimoolu

iMi w o d 4 rodmmsraiado

94 0

?an,

( )a

n do 0 5 r 5 n

+ 2y)1째

( )~

( 2 ~ ) ~

wodd r + 1 ~ o s m m a o i u R o ( r ) ( 2 ~ 2 ) ~ - ~ (= - ) ~( r ) (29-")(~8-31 2

~ ~q ~ U I L ~ Z U P I U L ~ ~ ~ L ~ ~

....mao+

.,-ioffs n i s n a a o ~ ~ .I ,~. o Y L . ~ ~ ( A v o J u I ~ w.-~ " L L ~ ~ ~ ~ ~ . b d ~ ~ f l $ 3 d f i i ~ i ~ a a o d ~82E1dl~~dtb fi~~~A'~~a I. Iuurniurydisnir 2 A 1 n$s fio.iiraunimaaosdu ~wnralu1souonI~<iwa~wa'dorriia4u %nGi ((MU?, %a), ($2, rYou), (6ou, Gi?) M?O (Coer, nYOu) 2. ninfinniYao'Tlw 2 Ka oiniYa5nw 4 Ka A B C bar D iirilun~~naaas~uiw~rmui~n ~on~6dioz~$~agn'13a AB, AC, AD, BC, BD ~ 3 CD e A

uau~ilan~da ~ ~ r m u o s w a ~ i u ~ r l d u ~ d ~ d ~ ~ n u aRindisrdu ~o~nl~na~os~~ 1. Iaurciiuq(si1s5~ 2 Ku 1 n%

In' H ununriiti? uar

T ununriiriou or16

rr~ub~na~dba S = {HH, HT, TH, TT) 2. nia~hnKao'nus2 Ka iainKa0"nw~4 6a A B C ~rarD ibgu~?laa~dlrS = {AB, AC, AD, BC, BD, CD)

f i i u % u b q a ~ ~ d q ~ o s n i 3 n ~ n o s ~ u $N ~ u ini ~ n d&iiidiu?u~tl~mr&~1:d~s~uam?rn.rd

b~a:~l:rGuiiiuqu~Dn~~dq un:

oiisAlilu~tl?nird6au

~ n ~ ~ n i r h n i n u i f i i n m i u a t n ~ 6 r l f i i a aoau. 6rls'~~~~

116 - .

a d s n ~ n q r n ~ El d ua: E, II b ~ ~ n i s d a $u1 uEl uar 4 ~9u~mqnim&o~ ~ n q m ~ dE~ d n E, = 0

In'

S ~iluumGlaadaua: E ~9u~nqmsd

n a u ~ i i ~ u u d w a ~ ~ n q nEi s dduuunuhu

C dom?m~diii(rlanou~au~ui~nf!of'Luu"~~o~~~

ada S ud!doflum?ni.id E

5%ua:~nqmr& ih'hu%u~Tlaa~drlr S i?aui& N &~nudau~Bnuda:&~~nia~iia~u!S~vi ~Duh~a6rlaoc S i i a ~ i hR iiq ~ a i u ~ i i l ~ z ~ ~ ~ d ~ : ~ i iEa ~duuunuii'au n q n i s d P(E) b u d

1. n~iuli~a:~iluao~myni3d E b7 seddi&ud 0 BJ 1 ~ u ~ o0 I P(E) 2. P(S) = 1

uwr

P(0) = 0

51 E, ~wr E, r i l u r n y n i ~ d ~ o ~ r n y narlk i~d

5-1El

5. ni

URZ

P(E1 U 4) = P(E1) P ( G ) - P(E1 n E2) E, ~ i l u ~ ~ y n i ~ d ~ orln' ~ d r ~ ~ i ~ u 5 ~

P(E1 u 4) = P(E1) E rilurnqni~a7a9 arlk P(E)

-4

?a911

+ P(E)

+ P(E2)

= 1

~ ~ s ~ ~ u ~ u ~ ~ ~ ~ B J ~ " I 3I YxI Q Iâ&#x201A;Ź I R~~ I ~ M IB ~ ~R ~ L~ $ J~15 M ~WABR RL ~ ~ ~ ' ~ ~ ~

RJUW

P(&)

5 N

= 5

10 x 9 -. 2!

45 - 15x14~13 91

~ n . r ~ n i . r ~ i a 7 ~ n u i ~ i a ~ % ' ~ b ~aoau. :~&a~ia~s"~

118

oin

P(E3)+ P(E,') = 1

P(E3) = 1 - P(E,')

ein (I) or'bK P(E,') =

ein

P(El U 4) = P(E1)

24 -

67 24 Q:GM P(E,) = 1 - - - 91 91 Y

+ P ( 4 ) - P(E1 n E2)

r. o ~ ~ l h ~ a ~ ? f l ~{a,n b) & ~uos~m ra I

1.1

1.2 a w

- bl

)a-bl

buff

= 5

5 5

2. uunnuu 12 nu lu4iuau&ZluCnfiuunQs 5 nu o ~ n i P i ~ a ~ ? ~ o ~ C m u n a u12 " nnu ~ ub~u~ds 2.1

rinl5uunQk.1 5 nu fidnKu~auo

2.2

lidu'nfiuun~s2 nu ImtuSmiitb

2.3

~ ~ ~ l ~ ~ ~ f 2i u ~~IO~U%IU . a 7 t LLR~AblFtflR l U ~ W U J % ~ ~ ~ L I U3M @ & . & O F ? W

.ld-ad

3. unaawaz&adidnl~rnou$i?uCai?nw~ 5 Ca i.raulJdidsbsnaiuvru7u6mul.a'Ka~nw.roin~a {A, B, C, D, E, F, G,H,I, J) Cai?nw~~uuoiardi~~andisiiunum

9he'nw A, B, C, D,E,F P~B$M~IUWJ"J 1, dllrfldd 3 ~?~dlLLllddC( 4 Ldllk4 4. diuau$~z~4is 3000 &I 8000 dddiuau'bauCfudarwanuaasaudisn'~ 5. ~);1~1Fi791091-11+ 2.2! + 3.3! + .-.+ nen! 6muA n ~ ~ ~ d l u a u ~ u

d U

7. osni~iuau~ani~d~~u~iuau~~uuanuos 1040 ha:: 2030

8. o s n i d i u a u ~ i r n ~ ~ i ~ u ~ i u a u ~ ~ nu uda sn~~qousw pru~os ~ 3 8.1

n=210

8.2

n=630

8.3

n = 151200

9. otnidiuau3Pdae~~onCn8uu 9 nu oinrinhuu 15 nuddxnou&ulin0uu~iu 10 nu CnfiuunQs

do.

. l

5 nu budn~u&fiuuddonu~orGosuun~~uu~Q~ 3 nu a

10. ubnioai9 10 aaamnusuna

d w Yo os~~diuau?'iC(oe-J"m~~nCnfiuu 7 nu ~suu~iiif6nun~nioa1sudarKa

lrioQ3milb 11. dnios 8 nia~aishusduun?at~idiuaui~donl,~nuoadis~u 5 an IdasninsIaudusiarniosld

~ntmalAiCius1 ;nuamdot.iiso~Gniiu 12. nquuosUn6uu 20 nu d ~ n u r i n f i u3 ~nu ~ ~g i~, g2,g3

uarlinfiuumu 4 nu b i , b2, b3, b4

o i h u asnidiuaui~a:41inhuu~ilu 2 una 7 a r 10 nu budrimfiuun~sgl, g2, g3 o~unanlii ~aarCnfiuu%iub l , b2, b3, b4 oaunangs 13. ~ ~ n i d i u a u i ~ a z 4 d n 7% nu i u ua:dnnQs

Gnaiu 3 nu

2 nu tufiusunaImud~iinne~n~ 2 nu;n&uGau

28. iaonnu 2 nu ainrnlinnui 6

a~ninaiuliioni3uuo~~n~n13dA

28.1

nundaadnubflu~iGn.r.ruifi

28.2

nunddbilusiu~nnun&bflun$~

nu~d~od%~b~urn~n~~uin'u 29. Iumr~i3onnxuni.r3 nu o ~ n g i u4 nu ua:nfJ~ 5 nu ~ ~ a i n ~ i u l i ~ a : ~ 9 d a : l d i n ~ a u n i ~ d ~ ~ u n ~ ~ oeild$ou 2 nu 30. y'aaeuau'ytshd 15 go bY~ionr5-1 10 $8 Iaufiinualfi~Qadi4 Sooin 5 Sobnn ~~nnun<ar51 ~oaoun.ru10 $8 a~win~iu~icr~9ud~"u1a~.iliSo 1 ii;l$o 4 31. n i o ~ l u n h u ~ g n u n a h 4i ? ;n uar'lh 3 i n n r i n b d a o ~ u ~ g n u n a l i3? gn ua:l(l~ 5 gn 4.4. ntugnuoa 1 gnsinnio~lunmu~ua:8n I ;nsinniodludao~ ~~mn~iuti,~nlfldn~uldignuoa~dii 2 gn 32. ~ n i u r d [ r i ~ d u ~ ~ ~:I~duubfiuquii?~on~d~n?uqa:#uw$ips'~~flu o"und~ 3 ifiiro~m&unriiriou <iIuu~n3u~&iouiiunoagn~9ii~du~a~~ 1 gn a~nin~iuriion9uCt 32.1 ~ n 3 u ~ ~ n ~ i n i u a e ~ n ~ d 1 $ u ~ ~ u ~ l u7nnic i ~ l1~ nu w f ~s ' o u + r uda: 32.2 rniu~g4~~1~ini 2 nA uarpbawuuaunnd 2 n%IunturnsIuuwiou7nir 2 33. ~ufl:nfi\undddgnm 20 gn ~ a u ~ g n u o a h i5.a gn 'mled 4 in %4u? 6 an uaduad 5 gn 61n t~~nuoanba:: 1 gn bmtuuri?liGunln~lIfZnai s ~ n i n ? i u l i i a n 9 u ~ ~ tow;nd u ~ n u1 ~ilu A mmio'nwi3o~ua:niiuanuoagnd 2 ~ ~ d d u ~ n i s l u a ~ 34. I u n i ~ a v i u5 nu uatn4.s 4 nu ~bi&~ouT'imau s~~imiuliianiluC( 34.1 il$d~& 4 ou~36aiii.4 34.2 l r i Q n e ~ a a ~ ~ u ~ a & ~ a i % 35. In' A ua: B ~ i l u ~ n ~ n i 7r dn~ul&u~u~~aada~du?n'uImud a 28.3

0 0 LLAZ

0 0 0 0

.. .

.*= A

,. **-

. --- E= *

.**

AB' =

1 0 -2-0 1

3 2 -1 - 2 3

1 .

-4

5 2 1 0

. r

94 0

aam

~dssoin A rilu~un"s.a"vuim3 x 2 uaz B r3u~xln^sndvuiq2 x 3

~ s $ u I i u a u ~ $ n mA rYiiT1uIiuaur~na"~os B rba: iiuauw$nvos B ~viiT1ufii~auunavosA d'

r w m r ~ $ u r n u ? ~ nAB ~ ? ua: BA k h i a ~ d ~ g u asu r

2 2

A B =

had14

6.8

6mua A =

2 -4

8 -6

8 - 1 5

1 3 1 2 2 3

1 : :I

10 -5

ua: B =

I 1 -1

PrWr AB uar BA

da i Z i i idora~n~unind A ua: B r i l u e n i n r ~ ~ r m u u u i2n x 2 iiiiurnui~nniAB ua: BA I.' 4.-

RJU

~ ~ ~ d n ~ ~ d i r i i n u i n i a m f ~ ~ ~ t ~ aoau. riimni~mi~~~

132

~ i n k o l i 6.9 i ~ o:&mduii

(AB)C = A(BC)

rh~~'~n!n~wiunddldif

niruiu 6.11

f i A ua: B ~ ~ u ' u n ' 3 n 8 ~ ~n7xs ln 4 4 AB = BA = I, w h n B i? run3ndwnu'uslo.1 A ~!uuunu B 6au A-I &G!a B = A-I ttadun A 41

1 $aeeiid 6.14

i q ~ $ ~ 6.6 w

AIMUVI A

0 -1

-2

uar B =

0 1 1 1 1

0 0 1

FIJ~HVIJT~I det(AB)

~xlw?n$ A

x 2 ~;~ti~~n?ndflnfl~nwo~~a det(A) = 0 uar

YUIR 2

det(A)det(B)

~nr~nirdiim^3nui~i1ai~a:n~(91~71m~~1~^a noau.

136

I u y l i n n i ~qudiu tu~ariniismri7aii~ni3~n~naiu~uw'~~~i~u61tu~d~un^rni 0-1 ~ ' i m ~ o i ~ m i n a i u G uruc A x B mu8zM' A =

{(q,

$, ..., h) Lag B

r i a A rrac B

= {b,, b,,

L~ML'IIB~~ITY~

..., b,)

runind 0 - 1 roonaiuihik.8 r r 4 e u ~ n u i uM, = [midm,. iiuiulmu

@%ad196.16

%M' A

= ( 1 , 2, 3 ) ) B = ( 1 , 2) ua:: r rflunaiu~u~u6ain A 1d B

auu8'lC r = { ( a , b) E A x B

< b)

.ilor%iuu~nu r 6auwn3nd 0 - 1 M,

gae419 6.17

A =

{(q,

4,%) ua:: B = {b,, b,, b,, b4, b5)

o~winaiu~u$ua" r ~uo

0 1 0 0 0

M,=

1 0 1 1 0 1 0 1 0 1

~uoosinr danouhudoiGu (4, b,) E r

r = {(R>4 1 7

wuuGlM' A =

{(q,

($7

&uu!K

bl), (az, b3), (a2, b4), (%, bl), (4, b3), (%, b5))

..., a,) d

Lag r ~Sunaiu~uw'wduu A

r ~~uu>nzGouiiiaruo ( 4 ,4) E r

$uta

. I

ruo mi, = 1

oel&ii

?n i = 1, 2, 3 , ...,n

nifio7.r~einauiBnansrll~nP3nB o - 1 Mr ~ A j ~ e ! i Y i i

i~~d

Ifir~mrdin~nui~ciaaf~~~3rfi$a~ci1mf~a~% asau.

138

FIJMI Pa

aanl

A V B LLa:

AAB

einuniluiu 6.14 d d i i

AVB =

ov1 1vo ov1 1v1 1vo 1VO

1 1 1

1vo 1vo ovo ovo O V l O V l IVO

1 V l OVl

aani

~inunQwu6.18 o:16 1 1 1

M,,

M,.8 Ma

0 1 1 0 0 0

1. filMM@ A = [-2

unt B

1.1

1.2

A - B uat B

1.3

5 A unt -3B AB Lbat BA

1.4

3 -11

Lblt

B = [4 -4

+A -

93Mli72103 x, y, z L b l t w 8lTIAa.lfll3

x-y

2w+z

83MI

11. o ~ ~ p l i n q w d m d7 isialdd ~ 11.1 wqwg~n6.2 $8 (ii), (iii), (v), (viz) un:: (viii)

11.2

n q ~ i u n6.3 $a (iii) un:

(v)

ssni A

v B, A A B uar A 8 B

17. w l "r = { ( l , 2 ) ,( 1 , 3 ) , (2931, (3,319 ( 3 , 2 ) ) s = { ( 1 , 3 ) ,(2, 2)) ( 2 , 3 ) , (3,111 rflunaiu&W'Pahur~~ilA = { I , 2 , 3 ) BJMI M,, M,, M,, us: M,, 18. rfiv\u@ A = ( 1 , 2, 3, 4)) B = {a, b, c, d) rrar C = ( 5 , 6 , 7 , 8 ) bi' r = { ( I , a), (1, b), (2, c), (3, a), (3, d ) ) rfluna~uiiuw'uh~nA \dCs B

bra: s = {(a, 5 ) ) (b, 5 ) ) (c, 7)) (c, 8 ) ) r8unaiuQurJ'Mhin B

Mr, Ms, Msr 19. 1w" r uar s rflumaiu~uw'uhub%mA dunu63ubuw3nd 0 - 1 QJMl

0 1 0 Mr=

1 1 1 1 0 0

0 1 0 uar M , =

0 1 1 1 1 1

{sf

&a0419 1

~i~ualM'~onnw~u~n+~ilu~'1la~e~41uau~ffu b ' 7 N(x) unu " X ~flu4iuau~h.uanu~:gu& E(X) L L ~ M"x ~Uu4iuau$) O(x) unu ~ilu41wuGi" UQ:: P(x) unu '(x ~ilu4itb?u~a~it" lLx

mmaiaoduu~anaiud~Iddlu~dd~z'Iun~~~n~d

Ca~aiu (I)

dazI~~i3~a"nwns"

iliuau~iiud~Ju41uaui

(4)

34E(x)I ~iuaudu~ndiuau~~u4i~a~~a~04iua~~ VX[E(X)v O(x))1 ~iuau~awiryn41u7u~~uQiuau~iuuan~~at~ud ~ . [ P ( x ) N(x)c)l 41uau~aw1:C(1~u41u~u~dci1~ilu 2 b'dE(x)A P(x))+ x = 21

(5) (6)

~~IW~MLOWIZL#UJBIM?ML~L~~LYIIMUYIL~U~IU~U~ 3!4E(x)A P(x)] 1li~iu~w~~uW~~u~iuau~un"a4iua~~ V Xf ~ [ N~E~( X0+ ) z ~o(x)] ~u~i~au6

(2)

(3)

t: d a

fiinualMionnw~uw'n+~ilu~.aa~as4iuau~iiu

IVY P(x, y, z)

unu "xg

= 2,

~mman~#uulmudaz~unnfi~~i~afdo~d&u~d~oadaz~un~~~n~d

cchvr%qn d.s~~u~~qGnwns" V ~ =X 0

Gaaaiu

rii x

= 0

-+ Vy[P(x,y,

uRI? xg = x hn?uyn pl' x)]]

(2)

~ R ~ I U

"hn?u~nx, 61 xy = x hn?u?n y uRY? x = 0"

vxpy[P(x)y, x)] -, x = 01 k~nai~d~zlunCqn"nwd VxVy[P(x,y, X) + x

¶Jszq?l

(3)

aswi

ucnnd~~oind~dunCryKnwd

b'@y[P(x, y, x)] + x = 01 Luaaoinwo x = 1 uaz y = 1 mm~&iid~zlucl CqA d Vx'dy[P(x, y, x) + x = 01 ozdh~llumfi9 ~ w d ~ ~ ~ z l u c l ~ ~ ~ n w d V@g[P(x, y, x)] + x = 01 oz$d~~iluo?~ ~a$url~zlunnda~[~~~f&ximui~n ~4uuunu &audazlun~yn'nwdVH~[P(X,y, X) -+ x = 01 16 Gma7u u i i ~ x y * ~y dC f x + o " d a z h a a ' q a ' n d V43y[-P(x, y, x)] -+ -(x = O)]

-3xVyVz[P(x,Y,z)] * VX[-VYVZ[P(X,Y,~)11 vay[-Vz[P(x, 9,zlll * Vd%z[-P(x, Y,z111

d.r:waM'd~~uiis ;a'li

- - V d y 3 ~ x z = y]

* 34-Vy3z[x

= y]]

3dy[--3z[x z = y]] 3dyVz[-(x+ z = y)] 3dyQz[x+ z

b o ~ t u 2 ~ u u u A ~ + i t o ~kulodo unii x ddiuinnii y d~worf3dyVz[x

y] onilus?~

uonoinf ~do~si.mr7ds:woM' uatd.r:wosa"

VxP(x) v Q(x) Qx[P(x)v Q(41

o:duiid3woM"~~~o~'~xi~ilud3:~oM"~~ua~u ud VxP(x) V Q(x) biludawalh~:clafk

pxP(x)] V Q(x) ~ d o ~ nVnX ~ ( Xv) Q(x) gad? x aoa P(x) ~*ii~~d~n~irillri?u~a~LiJrl^~uim V a w d n i t ~ dx~ am Q(x) LOU& ud.m3m

~daI$~+qIoi~u&m o i a d iclu x d d u k dr3~17:'1~'~lu& L L JK~&!K ~

~ x P ( x ) ]v Q(y) fioisrui~uvfiwosa~uaSiu o G !: vx[P(x)v

Q(dl

Q(Y)I

vxP(x) v Q(Y)

* vxP(x) A Q(Y) 34P(x) :)V Q(y)l * 3xP(x) V Q(Y) 34P(x) A Q(Y)] * gxP(x) A Q(Y)

vx[P(x)A

~du

a:nonh&

{a,} i~ labaa. t Y i u s i a : w o ~ l u i i ~ a o a n i o ~ ~ ~ a i u ~ u ~ u fi&odi~do!du a'~~o~~iia~u

Saodw 3

~ ~ d u u m i u ~ ~ w ' ~ d ~ ? ~ u b i i a r o ~ f i u F j i n ~ u ~ ~ ~ ' L ~2% % u(aonbdsnu~u) ~ni~'Lflaon~d~~ae

osinnnuiidonm n S s e d D u i n r i i i d o d u ~ i n ~ ? i ? o20,000 ~ u uin b s IrinouQuainCq5hao wiou$~ni~uhnluY"~ir~donm;jA 3

i s d l 18 a,, ~ i l u d i ~ ? u n ~ d( i~u3 ~ ~6:)a n ~ o ~ 4 a n d IRU# q, = 0, q = 1 w: a,

gaud 5 smadls:\6 L

uuna

'.,

= 1

% = 2 = a , + % , a4 = 3 = a,+%,

a,, = a,,-1 - a,,-,

5 = 5 = a4+Q,

...

h ~ ? on 2 2

~ ~ ~ i u w i ~ m ~ ~ ~ r i a n d ~ m u i ~ n w t a1~ 6uyn7 n ~ ~ b i~ nl onui$uiGdo~ d a d 1 wial6 1 6% lad 2 U 2 6u~iluduad6au~ ~ u i i ~ d o uni wPd6 n 6u d

s P ~ d u u n a i u ~ ~ ~ f ~ ? u u ~ ~ @ ~ u 3 w nnu ~iiouum ~:w^a~~~1~du d '3sA-1 'bf a,, ~ i l u d i u ~wur l ~ n ~ n u a d ~ u n ~ ma G a ua,~ = 0 &a: = 1 oz16 a, = 1 + 2 = q + 2 , a, = 3 + 3 = 4 + 3 , ... ~ ~ i f u a,, = a,,-, + n, n 1 1 r

152

h~sni~~ir7?~ui~iawf~a:nfia~ia~f;a~^a aoau.

hod14 7

~ d ~ J " U l ~ ~ H B 1 ~ ~ o ' n ~ ~ 1 ~ ~ urn ~ JL ~M d~ ~Y 5 ~uin ~ buarlniuql ~ ~ l : b10~ uin " a ~ ~

~viiuu

o~duunaiu&w'ud~~uu~fiauos~iuau~"a~~1t~uoa~n"auqdu#~nua n urn d o

5k1uamudnni$ ~uu3iihuau~niuquuuiis 7 d!$;Id4iiia a4

asm

b '7

Ro 41uaui'muomn~uq(~9uBiuau;~i1n uin t-Yid~nfucy3 %3a a o 1,s ua: 10

i~ni&-4dud?"alK 3 ?~uaniisiiu -ad

an t 2

am 1 wuoa~niury 1 uin riou szafodu n - 1 uin ~&snuoa~~d4iuauamsau an-,

add

am 2 nuon~niury5 uin riou annfodu n - 5 um Asiosnuo~d~d~iuauamsau an,

a n ll

h: 2 '

am 3 nana~n"aury10 urn rinu armloPu n - 10 uin ddiosnunadsddiuauam~~u an-,o

~~~a:~n~ni~~llidi6Iafl1d1iiu~au s

a, = a,,-1

+ 4-. + a,+m

n a i u ~ w ' u f ~ ~ u u ~ ~ a ~ u ~ ~ ~ ~ d d ' La, u ~ d=k a f(an-J !d~o wa b a a u a i ~ ~ i ~ ~ m ~ a a ~ ~ i u a ~ d i ~ i ~ i u a u I u ~ - ~ u[ i ? ~ ~ n a i u ~ u i u ~ ~ i n ~ i ~ ~ d ~ i~wmn eB~ua ~~ a~ i~~ ou ~u ~~ ui ~~~u~ innr i i , m p ~ i ~ m ~ $a uluni3Cmhl3 dd'

~ u n u ~ ~ n n w n i ~ ~ ~ ~ w a ~ a a u ' I ~ u ~ . ab"unnai m i ~a~s~l unwi ~ d~ai" ~did ~ n9a iunaiulGu&7au ii~ na~u&~u~duu&auw~1Pi~ioup9~1 rii1ddou 9 ~uii~di6u~'uuae~#naiu$uni~'~\i w a3au~io w a p idoln' o~luwadvsa n

~ n ~ ~ n i ~ d i . n ? n u i m a a i a b a z n f i w maoau. ~a~~a~%

154

wduu a,, =

n2 +

~fiuu~~aau~odnaiu%W'ud~?uubfia~uwaM"Uo~ n

6anLi1~~si~ a, = 2 ~ , + , 3 ~ ~fiunaiuiuw'ud~?uu~fi~~nnw'~~ , usitii a, = 2~~ - 3 4 4 + n2 + I. u ~ ~ n a i u a * u ~ " u ~ ~ ~ u u ~ i i a ~ x i ~ f i u n a i u ~ u ~ u d ~ ~ u u b i i m ~ n bwne f i ,,a = 0 ua: %, = 0 u62 a, 0

n i m i w a ~ a a u u o ~ n a i u i u w ' u d ~ ~ u ~ ~ i i a ~ ~ ~ ~ ~ u ~ ~ n w ' u f ~ u d k~ z ~ozi~uinnii ~~~udin~Kao'~Ku CuGuyld~ la ufloi~minai~;TUUw'ua'unuKa u 4 = d

buo q , $, ..., C k â&#x201A;Ź P

Clh-1

0 lJlJ

$4-2

+ + ckan-k "*

5 k

auuGlfi;drsuuwa~aauGo an = F , r P unu a, = r" Sunaiu~u$ud~?~u~5aoz1K r" = c l F 1 + $F2+ ." + c ~ P - ~ r" ' r"-k ~136118@l$i?~ F-k

c ~ F ~+ F c ~ -T ~~- ~ P+- '.' ~ + c&~??-~+ c ~ P - ~

oZ16 Ik = c 1 F 1 + $F2+ ". + ck

yl3S

clT"l

... - Ck

$rl"2-

..................(1)

= 0

A u n a u n i ~ (1) ii 1 u n i ~ ~ n w m t r a ~ n a ~ u $ u n~ n~ vS~"~~a 3u ~n(1) i~ ~~o~~ ~ i f l ~ G a i ? T g i ~ ~ ~ a w a ~ a ~ ~ uo~naiuriuwid~3uutR.Aunnnco~rrunis (I) ii nnrinwut G J ~ an = F ~ ~ u w a ~ ~ a u u o ~ n ~ i u & w ' u d ~ ? u u ~ ? i a ~ ~ ~ b ~ b o n ~ u f b v " ufido~do d a : : h $r ~d~ u d i n ~ ~ a d'

' ~ ~ ~ ~ O J ~ ~ ~ I " ~ ~ ~ ~ . W ~ ~ J ~ ~ I U ~ U ~ U ~ L ~ U U L ~ ~ ~ M

~ & ~ ~ i n n n $ n w ~ u o ~ n a i u ~ ~ ~ u f ~ ? u u ~ ~ a ~ Q ~ ~ ~ ~ a n w ' ku fJuinnii ~u3~~th~~~ud Y yl~~nn~~uni~~~irm~ua~a~u~a~daz~un'unn~~~~~~undo~du

6iyluanaiuiuw"ud~~uu~iia

= 24-1

+ 3an-2

LUS

1, a,

~~~~i~a~aauaa~n~iuCuw'udLjuu~iia

&Eld'ld

filM~a~27u~~w"%db?~lbLfi@

+ 64-3

a, = 9, a, = 10, a, = 32 ss~iwa~aauuo~fiaiu%r'uW'ud~~uu~fim an = 74-2

L~EI

-& - q - q

=-2

q+2q+4q

= -1,

rIitrunbi'~dmu~~iuu ABC dihuuia a, b aa:: e niau &-$ 1 ,

-:,

-,.,t

~ n s ~ n i ~ d i s 7 ~ n u i n i a ~ a " k ~ : ~ ~aoau. ~~1167~i~~ii"

160

ki 'I S y

f, ,

ozk%i

R ua:

r Y

~ ~ u i n d ~ ~ n a u ~ ~ o u ~ o ~ u ~ z u u ~ ABC l u ~ do:\& mu~~dtlu ~

abc

I ~ ~ M ~ ~ @ ~ ~= J LMRU = UTS

(i)

yu~riin'uynyu~~da:yu~nin'u60 earn

dsi w

a w ~ ~ d u u ~ $ i $~- au i u 8mtnwduunu~iu~rhn'uao36iu ~ ~~lfanii~iud~:notl~uuom orlXl1

(i) (ii)

qui~u~riln*u a:

~~~ainnn~uuo~ld~~a.in~u~iuat~bd~~83~~1uu1z~i~~"T~~uu~

~nsani.sd1~3nui~i~lws"ua:n~~~1~1~1f~a~^n aoau.

162

f i i % u a l f i @ m u ~ n ~ sABCD u &huui?

a, b, e uat d nJau

%@ 7.6

~ h n u u a y urnviufia ~$uawda~n~a~ulrl$~do$msaC~$~u C

~ h T ; anvisda ~&ms~Aain9inquldwaa1nC~~$unuu~yu

t rJ

~ I J7 ILMRUU~UWY~ # ~ # ~ d ~ a ~ u u k u=c in?ia i x sia

dnu

~ ~ H ~ U U ~ I H HVIU~J U H ~ UH L M R U U ~ U R I ~ ~ ~ J ~ I ~ V U I ~ ~ ~ U I ~ J ~

a:k'-ii

r h u u ~ y u d ~ 4 4 k u ~ : ~ ~ ~

164

~ e 7 s a n i s d i s i ~ n u i ~ 1 i a m f ~ ~ ~ naeau. fim~[i~(m~~~

3 ;d 15 asnwu

~&waouasuos?~nwu = 2m &damnu = d 8uAasu~au = r ( -~ P) ~ .. Lie R ~3us"Prda~nwuasuon uw:: r ~3ufokiasnw~aslu I

I-;

1 2

!I w

r ~Bunaiuuiai~d ua:

h ~ i l u n ? i u a ~ ~ o ~ n . r ~unr n ~ r4ui~t lnu i ~ ~ t i u ~

oin

(2)

oin

d'd

= wun31u X

&RE

daguu ABCD ~ ~ u @ ~ ~ w a u u & j ? a

a i n y 24

b ~ ~ F l 1 l ~ ~ ~ l l 0 ~ ~ 5 t h l

&a AAOP w r AABC ~ ~ u ~ d m u m d u u ~ u a i n at?& $~%l

??n A@

3 ''

OP = A P - A 0 = 13~ 52 = 14.4

OP = I 2

+B e

A@ uar AB = AC

qr?&

a~niwunu.m~wun~o~~~rfin~i~~~yIauuWuW"~ ~ i o ~ i u r i r r r i n n10 i i ~ rjn uia 18 vjn uar iC~rfingd12 vjn d

isdl l i i y I u n l $ ~ . r : ~ m ~ ~ i u b a u @ ~ ~ABCD ~ a o u ~$4 u ~PO ~ &id 25 o i n y 1%AB = 18, BC = 10 uat PO = 12 ad& OR = 9 uar OQ = 5 ~iodainAPOR uat APOQ d~idmuaduuuuain OZ!~

+ OR2 122 + g2

PR2 = Po2 =

= 225

PR = 15 PQ2 = Po2 =

+ oQ2 122 + 52 = 169

PQ = 13

Y dr

2 d

ruosoin wunwan~nums.6IrtG-a = wun~iu wun~aGis

- 1 0

asni

Y d-

wunu2ui~uo~rlilu = ~ ~ i u u 1 3 1 , h ~ a uxpIu~ J

= (3 4.5 = 333 Y d

+ 5 + 6) x 18 Y

911Tl9~2 Y

u u h :

wpannuinmna 2 6iu = 2 x 16 = 32

~1:yg

Y d e h: WUYIFI?MJY\UFd

= 333

+ 32

nim~aa "

= 365

61137Jii7

~wm:iidin~~ni~sifaa: 2 uin

K~$U

%adw 7

~ I ~ I ~ ~ W U @= L ~ 365 M X~ 2 M =

730

UIn

C ~ ~ ~ ~ u n d ~ ; d n w n a : u a n d a i u # ~ ~ a ~ G i ~ ~ ~ u ; d n a ~ u ! u u i ~ ~iinjg , d i ~ ~20# ~vjn ~a~d~~u un:n2iuu12~awi:naana:uan 15 vja ~ ~ 7 ~ d I u i % ~ 308 a i ~ -tp~i~6qm ij OJMIR~IUUI-J u a ~ ~ ~ w i q u h a ~ ~ u a ~ ~ d ~ ~ ion ~mnasna:u

172

~ n . ~ n i $ d i ~ n u i ~ i r ~ ~ ~ b a ~roau. ~~fia~~ir~s

A C : CB = 2 : 3

oin q,

m1+ 77%

m+n

Yo =

ny, + my, m+n

Y - Yl = 4% - %) ~uniaha.r~dsi.wu x d9a (a,0) uazhunu y Cfga (0,6) Ao

htlancn

+ cl = o by+ cl = o

r h a ~ax+ by L ~ U W ~ax+ J

+ =o ~ l : m ~ ~ l n ~ Y ~ ~ bxu B i lay 3 ~+ o j = o 0~~1.67~cYbh6r149 ax+

a y + c, = 0

M ~ C I-bx+

Zl;i d dw:u:ma~inina

( q ,y,) !dF~r&m.r~ ax + by

= 0 &!Q:

+ %Y4 + + %-1Yn + %Y*) - (331+ %Yz + X4Y3 + ... + XnYn-1 + q~n)I

[(qvz + %Y3 2

ABCDEF ~ 0 u ~ d n n m b u 6 i u ~hv~i id 7 aA, B o/uuunw x us: P ~ 0 u q a d ~ n a i r h u CD

61?a A 9fii%

(2, 0) kg? ~ ~ n i % a ~ o r qPa

a- o

arm1 yumulu$nmnduu~w~viini~~u~: 120 orwi yu OAF = 60

DJRI

6-m AF = OA sec 60" = 4

OF = OA t a n 60' = 2 dosoin azuz BD ~ilu2 ~viiaeuz OF 3:u: BD = 4 3 fiih?a B L ~ W(6,O), 67%3a C ~ 9 % (8, 2 3:

LI:

6aohd 13

7s PQR

~il~llailn~iuuhu~piid P(3, d ~ 1) m ka: Q ( - l , 5 ) b~u~muam

a s n i h d p m u a l u u PQR d

A'd

5gri1 buoiain wun$mu~niuuriiulrii naiuoia PQ = d ( 3 - (-1))' &!:+I

&asd?d 15

J3 x 4

+ (1- 5)' J3

6 u d ~ d m n b n ~ uPQR n = -x 4

f i i ~ ~ u v A, l j ~B

LR:

(giu)'

(m)'= 8

mirnsndao

C !G%~bau (-1, 2), (2, 3) bba: (a, 5 )

9 s ~ i d 1a d~li%~~$um%s AB " u ~ l ~ c U b # ~ BC ~S9

BIIU~I~IJ

nisiigad marutisnirGpJ~3uiluaosaou~o mad1

o::~~iluam~iifliA = B uRj A i l u a ~ ~ G iA '~ = B

B ua: B

~ ~ n u n Q u2.1 i u o:l&irruiBnyn.ivo~ L

ma~d 2

o::unasiini A

LR::

A ~ilumi%nvnsB

gsuu oinuniiuiuvosFu~qao::!&ii A = B ~ u ~ i u e s ~ # u a i o::!6ii iu A = B + B oin (1) ua:: (2) o::!6ii

--,

B ...............(1) ...............(2)

A = B + [ A G B A B G A] B c A LL& A = B

1% x

L ~ ~ U I U I % - Iq~1V ~ I ~ m ~ a n n w g a ~% Gn$

XEAUB * XEA V XEB

X E B V X E A X E

BUA

V ~ X E A UrB X E B U ' A ]

k'tbtia

AU B = B U A

. , .fi.

X E A v X E B U C#, *-. XEAU(BUC)

(uwfiuiu 2.5 $a 1.)

(up?Quia~ 2.5 "iio 1.)

&&A Vx[xE ( A U B) U C

x E A U ( BU

(AU B) U C = A U ( B u

lw' x

L ~ ~ U I U I %q ~I VMI

x€AU(BnC)

L'II~LB% ~~W~~JG~~

X E AV X E B ~ C x E A v (X E B A x E

(XE

* .

C)]

A V X E B) A

A V x € C) (~uGniansroioril~ A (XE

X E A U BA X E A U C XE(AUB)~(AUC)

u ( B n C) +-+ x E ( A u B) n ( A u 01 ku#a A u ( B n C) = ( A u B) n ( A u C)

' t b

V ~ E XA

(up?Guia~2.5 $a 1.) (unQuiu2.5 "iio 2.) L

W V~ )

(uwGuiu 2.5 %a 1.) (uniiuiu 2.5 u"o 1.)

~ n s a n i r d i s i 1 n ~ i m n a ~ ~ ~ ~ n f i a ~ noau. ina~~wfi"a

nm6nd'

rnszfipsw"~~~~: ( i )( i )( i )( ) (

(i)

1% x X E

(

) uar (&)

~ ~ u a u i %qn1u~~n~onnwEuCwd l~ P

AU A

X E

X E A

A v

X E

6 4 ~A U~ A = A -4 1% x ~Bunui9nlaq ~ M L " I I ~ L o T ~ ~ w ~ J . N '2P ~~~

(uii)

9% x

bBunui9nIn B] %~~qa~onnw$u.N'nd 22

xEA-B

t ,

xEA A x@B

XEA

Vdx E A - B kuto A - B G A

x E A]

(xiii) nuulii A c B arfiasdii A n B = A sin ( x ) ~ d 6 - A i ~n B G A IWf7~ilA

C A

LLQZ

(k) ~ U <I A

A An B

qin (ii) ~1::lKji A {atu ni A B (xiv)

A nB

~ $ 2A

'Lfi x s9uauiQnZm q %us(nmsonnw~uw'nb P XEA-0

XEA A x$!0

ssd"Un2iu x 4 0 silu-a?a~aua X E A - l a tt X E A 6a;u tu6a A - 0 = A

( u n h i u 2.7) (nqwaun 2.6 (xvi)) (nqktfun 2.6 (v))

n?Atopf (i)

x ~9uauiBnImq Iusam~annwGuGniP

(ii) X E

AnA1

A A X E A'

X E

X E A A x$!A

d~~ftm~a~m~0~9

A! = 0

nlsliinlf

~DHC:1

mod2

I us:: A n B = 0 IUU%I B = A' 8d&unqk+iern 2.7 ii A U B = A U A t = 9 ~blz A n B = A n A ' = @ aeriprii, ri, A u B = I ua:: A n B = 0 L& B = A' I U U Z ~ I A u B = 9 at A n B = 0

~ z i a ~ d $1 ii B

= A; uga A

B = P ~ B = ( A u A') n B = ( A n B) u (A'

(nqwjun 2.6 (vi))

,. .

( n q ~ f p2.7 ~ ' (i)) ~~

n B)

(nqwaun 2.5 sruu3n1sn~~~1u)

u (At n B) ( A n A') u (A' n B) At n ( A u B)

= 0

(IUUG~M)

(wqw$un 2.7 (ii))

(nq~ijern2.5 ~ ~ e r 3 n i ~ n 3 z ~ i ~ )

Atn 9 = A'

(~u~ft31ta) (nqwiun 2.6 (vi))

n n m a d 1 uaemod 2 ~ n % q d ! K i i B = A'

nolarua A

= 9 uaz A

= 0

n l d i o 6 m s d p d ~ ~ w i(i) z hnitni~fl~~d~o~d~~ijooeij5%idn~iunQ~n'u (i)

IMY

x ~?lu~~i%nlm qI~~

XE(AUB)'

LOIIRL~~AW~~<~%

xg'AUB

-t

" ( X E AU B)

" ( X E A V Z E B)

* xg'A

xg'B

XEAI A X E B

XEAI~B'

~ j l f t a ( A U B)/ = A I B~

(ii)

auu5ilw' a 1 b ~ r i 4 i u a u ~ &m

A ad

b = am

~ u a s a i n bc = (am)c = a(mc) oar mc ~iflu41uau~Ga.1 &$u a 1 bc

(iii)

auu5ilw' a / b ua:

fl:iBiuau~Gu m as b = am

(iv)

&&A m

& a

ain b

la1 5 lallml = lam] = Ibl la1 5 lbl

auuGl3i a 1 b oa:

uuna

a1 c A

OZ~~IWULG~ m &a: n q s b = am UP: 4 Luasain b + c = am + an = a(m + n) s '

(u)

5 lml

c = an US: m n

L?~M~IU?UL~U

aI(b+c)

~wG(LMY a 1 ( b c) UW: a 1 b A ~lrd41uau~Zum L L R ~n %S b -I c = am

LM:

b = an

ruoscln c = ( b + c) - b = am - an = a(m - n) d

n l n l a o i o r o l i ~ ~ a l h 02u mou);o

~mfi1

~rFip&in ~ f l u 4 i u a u ~ a m m ~ n q omm.induu?~u~duapm"uod~iuau~aw?~ car A bulPhrAnir~dliud~~1~m~1iammm 2 M 1 61 n = 2 16% n ~Su41uau~aw1r Pud 2 ~ u u i i ~ i ~ n ~ ~ u a u ~ ~ u u a n1A uo ii~nnnmi-ii i n oga n ~ i l u 4 i u a u ~ a m r m 5 na muim~ii uu!~~~daapm21os~luaM~awi:

51 n rflufiiuau~atnli: wmai7gau' nsGA n ~ S u 4 i u a u d r t n o u~ : b i ' i ~ i Q i u a u ~ G au LLar b dflilw' n = ab laud 1

< a,b <

L u n ~ n n 1 a ~ d l i u ~ ~ s n ~ m ~ i2a m arlriimvfs f 3 ~ d a oar b ~ilu41uau~awirm3n ~ u ? s ~ L ! U U ! ~ U , S ~ ~ W ~ ~ M ~ ~ S ~ I M ~ ~ L Q ~ ~ :

... p,

~uuG2M"n = p,p2

= q1q2

q, bun pi ua: q, ~SuPiuau~awi-ynriii = I , 2, d

PI I P2 I I Pr, Q1 I q2 I I ~a~nl3bblar ~ j= l s L:I pi = qi ynrii i -

d bt6090lfl

-at!-il

1 qlq2---q,

PI = qi

8~\6<1

&ti& PI 2

h99?UU14

b: LWSILGMM

pl = ql

f i p 3 . . . p T = q2q3...q8

lu~~uo~~~uanimu13n~;-a~$iii p2 = q2, p3 = q3, $1 T < s 16 1 = q,.+1q,+2 ..- q8 d~b!h4!d\ai!~ $ii r

j = I, 2,

\u$itbo~s~ua~ramul3nfi~-au"ldil p, 5 ql

..., r ,

&tfra\6 r

16 1 = p,lp,2

...

A ... p, %~bg¶b!d\d'26

= s UQ:: p, = qi

qndi i = 1, 2,

..., r

..., s

nwfipd e:urianifl;ipd~Ju~o~nouRo mad 1

1oe~ipjl.1ii41um.d~ q

In' s

uar r dfiilfl a = bq + r

= { a - bx 1 z E z URL a - bx 2 0)

iaut o 5 r < lbl Z

U~~L~RYOJPI~~ULBU u

Lue:u~aaIi S r 0 aau rdwcin b * o ~ J $ M b > o M% b < o n r ~ ib > 0 c:\& b > 1 6-zu -1aIb 5 -la[ ui -1 a1 5 a $3;~ -la1 b 5 a $a&.b!& a - (-1 a[b) 2 0 I a - ( a ) 2 0 &$u a - b(-(al) E S tuta S s s* 0 nrrGil b < o mu~~oR~~fidudiao~~4u~n"u!&ii u, A 61 0 E S e:\aai~ q1 E Z %a 0 = a - bq,

9:4;efiauI%~<nnirCa$u6uG

&$u!& a = bql + o laud &u+a o:l& r = o 81 0 6 s &$u S G N

mad 2

o <o <

~bl

~ ~ G f e JB i ih u a u ~ 4q~ UR::r ~ r i ~ a d f i u a ~ ~ i i $ u A . / i i I ~ " a = b q + r Tau# o 5 r < lbl Zt?' q,, q2, r,, r2 E z BJTI~IM" TauA 0 5 r, < lbl a = bq,+r, a = b e r2 buCf 0 5 7-2 < 1 bl

ua: + k: L a 6duu bq, + r1 = bq2 + r2 U ~ R O r, - r2 = b(q2 - ql) 9:!&41 lr1-r21 = ~ b ~ ~ q l - q 2 ~ ........................(2) z udn~rl R r2 L J ~ P I ~ ~ ~ L ~ U A 0~ ~rill~ ZlblV I%daualn' II~

IT1

~21

< Ibl

I bll qt - q21 < I bl

- q21 <

ua@Jil 0 5 Iql w

RJMM

141- 421 = 0

s:1Gi1

ql = q2

uszsin b 1

uei Iql

o sekiii 1 ql - q21 <

- q21

Lara47u?uL4u

6~tus1n (2) 1Gi1 r1 = r2

hdu io i d41uautRu q us: r ~ f l u ~ d ~ i i s a ~ i i ~ u d f l 1 1 S a = b q + r b u d o 5 r < lbl

tll3s@lH'

= {ax

+ by 1 X , y

, A e ~ n na r

o trfs

kfhjl%?uLdu Us: t

o ~ j t us

ain~R'nn7$ag%<u# sdGiid S W

+ by >

ua:

sG N

~ ~ ~ I ~ ~ ~ I U ' O ~d ~ B ~ M ' L ~ M

a, Q ~ f l ~dl f= i a x + by Biolrlszuaa~iid bu~uas~outrimimiil~iid~i'tb~u~~u q ua: r kflilri ~ ~ u w u 4 i w u ~x4 bra: u

a = dq+r

Iwd o I

r < d

a = ( a x by)q r s z l M ~r = a ( l - xq) b(-yq) ~ 6 o ~ s i0n 5 r < d ua: d a u i 3 n i i t u ~ m l u ~ %Sa s:kKii

<J&

uan~ii r =

o kuta

= ( a , b)

d1 a

Y1pu"Lu~iun~~iiu~n'u~&ii d 1 b &ifu1& d LBU&MIT+~~J~OJa n'u b auui1G e ~iludiuqu~ituuqnd~ c 1 a us: e 1 b o:l&ii c 1 ( a x + by) ~ U A O d ~iluBY?~13ia~uinslm a {i~ b ~ ~ : ~ ~ ~ I W U L G2 ULA:: y QJ

6ajifu

d = S+b$

ndigod

d = ( a , b) unz m = [a, b] d

LsaoJain d 1 a u ~ : d 1 b

std4iuatb~~u x, y

d %J

a = dx st b = d y ab ez!Kii bx = ( d y ) x = ( d x ) y = a y = d ~ d o ~ s ibx n ~ i l u w q p r u ~bo ~us: a9 ~iluwypnruo~ a ab d4 a ab J - LSU<?~(U~~UIIDJ a ua: b uuoo m I d d sz15i1 m d 5 ab L r

MUFI o

[a, b](a, b) I ab

a n h ~ & < a i i m 1 ab

R~~~-~:G~IU?ULRUs

dflib?'

ab = m s

uwzoin b 1 m %&A!$

oeiidiuaw~8u t ?!fiiIG' m = bt

u~a~luiiuo~~iiuaiiu16di s1 b d. a

MUfla

"a:

u ~ a ~ iai = st

ab = bst

s1 a

nJuu

&~#o s 5 d

or16 ma

........................(2)

I [a, bl(a, b)

m (1) unr (2) qdbkh

ab = [a, b](a, b)

nidialf

ruoJoin m 1 ( a - a)

(ii)

wuu;l$

ktu a

a (mod m)

b (mod m)

%&A m 1 ( a - b) atii4iuaudu k dfill$ a - b = mk nw:a:uu

(iii)

b - a = m(-k)

tu4o m 1 ( b - a)

auufta$ a a b (mod m) uwr b

&tux b

c (mod m)

9:!6ii

mI ( a - b) ua: m I ( b - c ) d ord4iuaur8a.1k, n ~ f i i b 7a - b = mk uwr b - c = mn w

(iv)

asuu

&.dio

c = ( a - b) + ( b - c) = m(k + n)

~uuGl$ a a b (mod m) bra: c o:i4iuaur8u 4

&~#o a

k ua: n

~IIJ

a =

= d (mod m) b + mk uw: c

a + e = ( b + d) + m(k+ n) ac = ( b + mk)(d + mn) = bd

uwr

(v)

e (mod m)

HUNG%$ a

(mod m)

US:

= d

+ m(bn + dk + kmn)

bd (mod m)

b (mod m)

w"o~nimsra~ii an

bn (mod m) #and

+ mn

qn1iu7ur8uu?n n

~nu~%~~nm.rgd~ur~Jnsii,919[i~~Tam 1 auuGl$ ak I bk (mod m) d o k ~3udiuauduuan

a (mod m)

am$a

aka a bkb (mod m )

(iu) !&I

G~&A!RI a&'

= P'

(mod m)

~tilu~~nniagd~ub~s~Sii,~~is~arabn 1 !&ii an a bn (mod m) vy-1Biu2u~6uu2n n (ui)

auus%v!' ac 6 s uanrii

n7dpos"

14' 8

= (a, b)

bc (mod m)

( a - b)m

- I ( a - b)

bbar

( c , m) = d

-k d C

&RE

g = (b, r )

a m d = (a, b) add' d 1 a UR:: d 1 b G3Zu d 1 r a i n g = (b, r ) ad; d 1 g % u ~ i w a ~ ~ i i u ?ari inu g = ( b , r ) arT6' g 1 b uar g 1 uarain d = (a, b) astln' g 1 d && d = g

4.10 l6ii (a, b) = ( b , rI) = & Luasain r n l rn-, ns~6~!& (rnTl,rn) = rn L a uuna ( a , b) = rn

n i s l i a d a i n n q Jun d

( r l , r2) =

G6u g 1

...

= (rn-2, ~

~ - 1= ) ( ~ ~ - 1rn) ,

,pguu 4.13

I 81 a ~ i ~ u - i i i u a u ~ iu~a t rn ~ i J u i i u a u ~ i ~ u a n d u 1 n n1a 1114

(a, rn, =

am-a

ou~ao3aws z usqba m O TG ~M II

d n ~ a S i p d ruodoin e .

ax+ my

RJUM d

s z d 4 i u 2 u d u x ua:

(a, m) = 1

LMoJ3ln my

9 &YII~G 1 = ax+ my 11. ,.;

1 (modm)

~1:1&<7 ax

(mod m)

1 (mod m)

d.'ui% z ~i3uZunoSmoa a uo?la m q ~tldlmoi~(an.jnd~ua o ~uoqla m

RUU%I

~1zk'ii

RJWU

ax (mod m)

~ u o ~ o i(a, n m) = 1

ainnqw&m

u"o (vi) 18

1 (mod m)

x (mod m) d.'~fio 'ounofmoa a d d u ~ d ~ r i i u ~ ~ $ i ~ u ~ b n _ o ~ n ~ ~ o ~,q2d n " u ~ u u o ~ b a

r n f i q ~ l lLdu o J n n

..., n

1, 2,

d~qu-iiuauc~uIg~ilu yi a:

Mi%yi mi I q a j y j

njuu M,y,

= 1

qn i

= 1,2, ..., n

1 (mod mJ

4 (mod mi)

M, d U@RJ<I

mi r B u R i d o n o u

Mjajyj

M 1 4 h + M 2 s y 2 + . . . + Mnanyn

$uo;

x = Mlqyl

LuoJain (mi, mj) = 1 ~ u oi

&$u16 mi 1

0 (mod mi) 4 (modmi)

+ M 2 s a + - - - + Mnqyn . I

x = MI%&

(M,, mi)

s i n n q w a u n 4.12 \& Mjy iiE 1 (mod mi) dwabaaudua

gJih d

RJUU

~ u o ~ ~ i n ~i dra jz o:18ii

(mi, mj) = 1 ~ u oi E j

~ M ? U L L ~ A :i =

LWTIZXUU d

4.9

i = 1,2,..., n

~ ~ u w ~ ~ a a u ~ ~ ~ . r : u u(1) m~~n~~otb~

buuuu?ln$~unA

+ M2sy2+ ... + Mnqyn

do 49. U q = x

(mod m)

b ~ u w a ~ a a u ~ oi i~u~a:~u u n o u n ~ ~ o (1) ud

nmlgd mstfip6awi: 16' A (2)

= [aJ,

(i),(iu) ua: (ui)

B = [bi3] uar C = [cij] L Z I M L B J W P ~ ~ ~mY Ux I ~n

(A+B)+c

= [(az3fbz3)+cd

( f l w k n u 6.5)

+ (bZ3+ cZj)l

(iv)

= [ai3

(~uu3n13<~n~uvosni~'uan)

= A+(B+C)

(uwQaiu 6.5)

~ w a i t 1 1 -A = ( - l ) A = [-aid A

+ (-A)

= [a,

+ (-aij)]

= 101 = 0

'iv) nlm'anM'

m s t r i p d ~ a w i t ( i ) ,(ii) (i)

%ss" A

= [ad,

k = 1 , 2 ,..., r

LLRL

(iu)

B = [bjk] LLat C = [ c ~ ] Luo i = 1, 2 ,

uat ! = 1 , 2 ,..., s

..., m,

(ii)

'la A k =

niAgoPs" (i)

(ii)

[d,B = [b&] uaz C [cd 1,2 ,..., r uaz j = 1,2 ,..., n =

'Lw'A

(iii)

= [aJ

~ u ai = 1 , 2,..., m ua:: j = 1 , 2 ,..., n (uniiuiu 6.6)

(kA)T = [ka,AT

(iv)

= [kajil

(unQuiu 6.8)

= k[aJ = kAT

(uw$u'~u6.6) (uwik~iu6.8) d

A = [ag] ~ u oi

= 1,2

,..., m. &a:: j

= 1 , 2,..., n

(ATIT =

n-nw'#o&

Iw' A

= [ad

= [a3IT

(unftuiu 6.8)

= [ad

(unftuiu 6.8)

= A

(unfiuia 6.8)

roJa i, !. = 1, 2,

AI,, = [f: aU6d =

bijl

= A

I,A

[C &atd I

= [ad = A

..., m

&a: k, j = 1, 2,

..., n

(un$uiu 6.7)

1. AFUTHMEUM rechnen einst und heute in the Research Institute for Discrete Mathematics. Lennestrasse 2, Bonn, 1999.

2. H.S Hall and F.H. Stevens, A School Geometry part I-VI, Macmillan and Co.Ltd., London, 1955

3. Howard. Anton, Calculus (A New Horizon), 6th ed, John Wiley & Sons, Inc., New York, 1999. 4. http://www-history.mcs.st-andrews.ac.uk.

5. Johnonbaugh. Richard, Discrete Mathematics, 5th ed, Prentice Hall, 2001. 6. Redmond, Don. An Introduction to Number Theory, Marcel Dekker, Inc. New York, 1996. 7. h s e n . Kenneth H., Discrete Mathematics and Its Applications, 5th ed, McGraw-Hill , 2003.

* V

3 3!

@ IAI A E B A c B A $ B 0

AUB A n B A-B A' A x B r-I s o r

sac?

a'r b

a ais b a MI3 b

[.I

~ J ~ ~ ~ ~ ~ I W U L ~ U 2X ~ 59 I ~ ~ ~ ' U ~ J

( a , b) [a, bl

a mod m a

= b (mod m)

a f b (mod m)

AVB AAB

58 58