Z Z

: . - . - . - (.n ) - . -

.Z - I

Z -II Z -III

(Z ) II

Z I

: :

: .3 3Z

-3 -3Z . -3Z 3Z

: :

r Nq Z )r,q( 8 037 r + q81 =037 < r 81 :

: 63

: :

Z k : 3Z : -3Z

.Z k -3Z 3Z x 3Z = -3Z

3Z -3Z x Zk : k3 x x 3Z

k- = k )k3-(- =x k3=x k3- =x ' )k-(3- =x

x-3Z k Z k3-: x x-3Z

-k = k k-(3 =x') k3- =x x3Z k3=x

-3Z= 3Z

-3 3 :

037 81 81 037 01 04 037 =04x81+01

81

22 x23 = 63 E 63

}63,81,21,9,6,4,3,2,1{ =E .63 1 63 9

: * :

GcEbDa=A: 1 D G,E, a,b,c

)1+G()1+E()1+D(: A

: Z N -1 Z }.,3,2,1,0{ N

}..,3,2,1,0,1-,2-,3-,.{: ZN - }21,9,6,3{=A N -

3 N .1 N B - : *

Z @ @3 , v-

: -2 : a,b .b a b a . k=b k : *

.b a b/a b a :

9 8 27 27 8 9 x 8 = 27 .9/27 8/27: 27

:Z -3 : ( )

r Nq Z )r,q( b a b

: -4

Z +38IT -I

n

weN Y

... )AN (

=EMA Z

) (

MGRP

RETNE

RETNE

AN : EMMARGORP :

R B A Q : -II

AN : EMMARGORP : B,A tpmorP : :

A B

n O / I tpmorP : B ; A

AN : EMMARGORP : B,A tpmorP : C )B/A( tni ::

AB

C

Y MUN

)BA(tni C OTS

CxB-A

D

C x B -A Z

D otS

C = Q

O/I [PSID 1 HTAM dn2 QC ;

retnE

MGRP

2

RETNE

HTAM

5

RETNE

RETNE

MGRP3

AN : EMMARGORP : B,A tpmorP : C )B/A( tni : D C*B-A : D ;=Q psiD :

5 D=R O/IPSID ; 1 HTAM dn2RD

retnE

dn2 tiuQ

edom ; :

naelC : tiuQ

AHPLA :

: RETNE AN ; CEXE ; MGRP

? =A : RETNE( 3724 )

? =B RETNE ( 122 )

(1=Q 9 ) Q) (47=R ) ( R B A

MGRP

3

: -5 b a

rd0 b r+ qb=a qb rd0 b qb qb+bq+brd

qb )q+1(bq+brd b )q+1(b qb qb )q+1(bad:

: :

qb )q+1(bad b a .b a q

. : * 14 372 :1 372 = 14(6 + )72:

)1+6(14 < 372 )6(14 782 < 372 642

.14 782 642 52 -533 : 2 -533 = 52(-41 + )51 :

)41-(52 < 533- )41-1-(52 053- < 533- 573-

52 -053 -573 : -6

: x x 42 ) x

( 42 32 22 12 02

N : * }1{ 1

252 : :

252 2 7 x 3 x 2 = 252 : 252 621 2 :

36 3 (2+1()2+1()1+1 = )81 12 3 252 1 81 252

7 7 1

252 22 12 02: 2 23 13 03: 3 17 07: 7

7 3 7 3 2 - 3 2 7 -

02 x 12 =2 12 x 13 = 6 02 x 22 = 4 12 x 23 = 81

02 x 13 = 3 12 x 7 = 41 02 x 23 = 9 22 x 13 = 21 02 x 17 = 7 22 x 23 = 63 22 x 7 = 82

: 13 x 17 = 12

23 x 17 = 36

7 x 12 x 13 = 24 7 x 22 x 13 = 48

7 x 12 x 23 = 621 7 x 22 x 23 = 251

02 x 03 x 07 = 1 252

}252,621,48,36,24,63,82,12,81,41,21,9,7,4,3,2,1{= E :2 252

1

2 2

2 4

3 3 6 21

3 9 81 63 3

7 41 82 12 24 48

7 36 621 252

7

Z -II

:1 .7 68 821 :2 912 = b A

b < a

821 = 7(81 + )2 68 = 7(21 + )2

2 7 : *

7 68 821 7 821 ]7[68{

3 3 + )63(6 = 912 = b:

( b a )3 + q6 = a b < a

= n6 1 7 n3 63 = 3 = n7 3 7 n3 73 = 3 = n8 2 7 n3 83 = 3

1 03 63 (2 3 13 73

2 23 83 7 03 63

.7 23 83 13 73 : -1

b,a N 2 n n b a n b a

.n b a {a ]n[b : *

)b a( n b a : n 0 n b a n

2 n 01 { ]4[6 : 1 2 4 01 01 = 4 u 2 + 2

2 4 6 6 = 4 u 1 + 2 12 { ]8[11- :2

12 = 8 . 2 + 5 -11 = 8( - 2 + )5 : -2

2 n *N n Z r b a b { ]n[a a { ]n[b (1 a n a { ]n[0 (2 n < r 0 a { ]n[r (3

n a r 02 < 3 0 3 = r 32 { ]02[3:

: -3 .n )b a( a { ]n[b( 1

r + nq = b r + nq = a a { ]n[b : nq nq = b a )r + nq( )r + nq( = b a

n )b-a( n)q q( = b a 2n 0n n c b A( 2 a { ]n[b ( 3

( )a { ]n[c b { ]n[c @n>b{a @n>b{a ( 4

b+b{'a+a' @n>: b-b{'a-a '> @n

b.b{'a.a '> @n @n>pb{pa

*Np :

.n a b a { ]n[b :2 n a b a { ]n[c

n a c c{c abn@ >a { ]n[b :3

c{c aabbn@ > c bnqr r+qn=a:

2 c c bqnr 1 c c aqnr

(1) c c aanqqrr

) c cc bbnqqrr

2(

n c bb c aa

c{c aabbn@ > .

*Np pa { ]n[ pb a { ]n[ b: *N p

)n(A 1 = 0n )0n(A (1

a { ]n[ b 2 = 0n )0n(A

a { ]n[ b a { ]n[ b

)0n(A 2a { ]n[ 2b u{u aabbn@ > 2 = 0n

0n n )1+n(A )n(A (2 pa { ]n[ pb:

1+pa { ]n[ 1+pb a { ]n[ b pa { ]n[ pb :

u{u apabpbn@ > )1+n(A 1+pa { ]n[ 1+pb

*N n ))n(A(

: * . n

: * . n

: 42 { ]6[ 63 (142

663

6 6

4 { ]6[ 6 61 { ]21[ 4: ( 2

21 4 61 @21>2 { 4:

: }3,2,1,0{ n 7 n2

7 246 1 :

02 { ]7[ 1 02 = 1 : 12 { ]7[ 2 12 = 2 22 { ]7[ 4 22 = 4 32 { ]7[ 1 32 = 8

7 462 1 32 { ]7[ 1

n32 { ]7[ 1 n32 ]7[ n1{ 46 = 3(12 ) +1:

1 + n 3 = 46 1 1+)12(32 462 1

1 462 { ]7[ 1 12 x )12(32 1 462 { ]7[ 1 2 x 1 1 462 { ]7[ )1 2(

1 462 { ]7[ 1 1 7 1 462

: n 7 n2

n2 { ]7[ r 7 n2 .7 n2 r

Z

:10

.5 n2 n (1 : 5 (2

70023556 = c 4672273 = b 26532 = a

:20 11 58 (1 .11 018 (2 .11 20028 + 2 (3

:30

.7 n2 n (1 3 n (2 .7 1 + n2 + n22

:40

: n

{ 1+n25 + 1+n23 ]4[ 0

:10 5 n2 (1

02 { ]5[ 1 12 { ]5[ 2 22 { ]5[ 4 32 { ]5[ 3

k42 { ]5[ 1 2 k42 { ]5[ 1 }3,2,1,0{ +k42 { ]5[ 2

{ @ > : {

21[5]215

0

k4

k42 { ]5[ 1 { @ >

{22[5]

2151

k4

1+k42 { ]5[ 2

{ @ >{

24[5]215

2

k4

2+k42 { ]5[ 4 { @ >

{23[5]

2153

k4

3+k42 { ]5[ 3

:

n k4 1 + k4 2 + k4 3 + k4 5 n2 1 2 4 3

5 cb a (2 26532 = a x

2 + k4 2653 = 098(4 + )2 26532 { ]5[ 2+)098(42

a { ]5[ 2+k44 a { ]5[ 4 4 5 a

5 2273 = b x 2273 = 447(5 + )2 2273 { ]5[ 2 k4 467 = 4(191 + )0 4672273 { ]5[ )191(42

b { ]5[ 1 1 5 b

5 70023556 = c x 3556 = 5(0131 + )3 3556 { ]5[ 3

3 { ]5[ 2- 3556 { ]5[ 2-

70023556 { ]5[ 7002)2-( 70023556 { ]5[ 7002)2-( 7002

3 + k4 7002 = 4(105 + )3: 70023556 { ]5[ 3+k42-

c { ]5[ )3(- c { ]5[ 2 .2 5 c

:20 11 58 (1

08 { ]11[ 1 18 { ]11[ 8 28 { ]11[ 9 38 { ]11[ 6 48 { ]11[ 4

58 { ]11[ 01 01 11 58

11 018 (2 01 { ]11[ 1- 58 { ]11[ 01 2 58 { ]11[ 1-

018 { ]11[ 1 2)58( { ]11[ 2)1-( Nk k018 { ]11[ 1 2 + 20028 { ]11[ 0 (3

2 + k01 2002 = 01(002 + )2 28 x )002(018 = 20028

20028 { ]11[ 28 x )002(018 20028 { ]11[ 9 x 1

20028 { ]11[ 9 2 + 20028 { ]11[ 2 + 9

2 + 20028 { ]11[ 0

:30 7 n2 (1 02 { ]7[ 1 12 { ]7[ 2 22 { ]7[ 4 32 { ]7[ 1 Nk k32 { ]7[ 1

2 k32 { ]7[1 }2,1,0{ = +k32 { ]7[2

:

3@ >0

7 1 2

]7[1 2

{ k{

k32 { ]7[ 1 3@ >

1

7 1 2

]7[2 2

{ k{

1+k32 { ]7[ 2

3@ >2

7 1 2

]7[4 2

{ k{

2+k32 { ]7[ 4 7 n2

n k3 1 + k3 2 + k3

7 n2 1 2 4 k3 n 3 n( 2

2 + k3 = n 1+ k3 = n 1 + n2 + n22 = nA

1 + k3 = n

nA { ]7[ 1 + 1+k32 + )1+k3(22 ( ) nA { ]7[ 1 + 1+k32 + 2+)k3(22

nA { ]7[ 1 + 12 + 22 nA { ]7[ 1 + 2 + 4

nA { ]7[ 0

2 + k3 = n nA { ]7[ 1 + 2+k32 + )2+k3(22 nA { ]7[ 1 + 2+k32 + 4+)k3(22

nA { ]7[ 1 + 2+k32 + 12 x 32 x)k3(22 nA { ]7[ 1 + 4 + 2 x 1 x 1

nA { ]7[0 : nA { ]7[ 7 nA { ]7[ 0 k3 n

: :40

1+n25 + 1+n23 { ]4[ 0 1+n23 { ]4[ 1+n2)1-( 3 { ]4[ 1-

1+n23 { ]4[ 1- 1+n2 1+n25 { ]4[ 1+n2)1( 5 { ]4[ 1:

1+n25 { ]4[ 1 ]4[ )1+1-({1+n25 + 1+n23

]4[ 0{1+n25 + 1+n23

egatpyrC : III

. x

b + xa = y y ( 72 0 ) . b a

. )b,a( :

11 01 9 8 7 6 5 4 3 2 1 0

32 22 12 02 91 81 71 61 51 41 31 21

72 62 52 42 : *

b + xa = y )b,a( x y) x

( b + xa = y

( 82 ) x ( 62 )

: )5,3( :

: x 5 = b 3 = a )5,3(

5 + x3 = y: 82 x . -

22 (1 ]82[22{5+x3

]82[71{x3 x { ]82[51 51

4 : (2 x 3 { ]82[72 : 5 + x3 { ]82[4

x { ]82[9 9

5 + x3 { ]82[41 41 (3 x3 { ]82> 9

x { ]82[3 3

: : x

: 0

" : " )4,5( :2

: 4 + x5 = y y x

. :0 (1

y { ]82[4 + )0(5 y { ]82[4

4

11 (2 y { ]82[4 + )11(5

y { ]82[3 3

(3 y { ]82[4 + )51(5

y { ]82[32 32

: (4

y { ]82[4 + )11(5 @82>4+55{y

@82>95{y @82>3{y

: (5 y { ]82[4 + )21(5

y { ]82[8 8

: :

b + xa = y :

. x: x . y: x

: . 8002

0991 1302 - 563 663

001 4 ...( 4202 6991 0891) 004

.( 0042 0002) ( 0012 0091. )

: .1302 8002 (1 563 { ]7[1 ( (2

663 { ]7[2 2102 0102 (

. .1302 (3 0991 (4

:

:1302 8002 (1 8202 4202 0202 6102 2102 8002

1 7 563 ( (2 2 7 663

663 { ]7[2 563 { ]7[1 0102 (

0102 8002 9002 8002:

x 7 x 7 : (563 u +) 663(u )

8002 .6 0

V J REM RAM L D S 0 6 5 4 3 2 1

x { ]7[)563(1 +)663(1:

x { ]7[)1(1 +)2(1 x { ]7[3

3 0102

2102 :

4 2102 8002 x { ]7[ )563(2 + )663(2

x { ]7[ )1(2 + )2(2 x { ]7[ 2 + 4

x { ]7[6 . 202 6

1302 (3 71 6 1302 8002 32:

x { ]7[ )563(71 + )663(6 x { ]7[ )1(71 + )2(6 x { ]7[71 + 21 x { ]7[1

1 . 1302

.0991 (4 . 31 5 0991 8002 81 x { ]7[ )563(31 + )663(5 x { ]7[ )1(31 + )2(5

x { ]7[ 31 + 01 x { ]7[2

) 8002 (

: V J reM raM L D S 0 1 2 3 4 5 6 x { ]7[2 :

. 0991 2 . : *

. : " "

( ) : 9 0 21 )serrab sedoC(

. :

R 1C 2C 3C 4C 5C 6C 7C 8C 9C 01C 11C 21C edoc ud lc al R

)edoc ud serffihc sel( ... 2C 1C

... 5C 3C 1C

... 6C 4C 2C R

[ 01 ]0 R+ 3x( (+) ) x

Y R + y + x3 { ]01[0

7=5C 4 = 4C0 = 3C 3 = 2C 1 = 1C 6 = R : 2=01C 0 = 9C0 = 8C 0 = 7C 2 = 6C

3=21C 2 = 11C

. (1 : (2

R 6 9 0 5 0 6 3 1 2 0 3 9 : (3

R c 7 d 0 4 1 5 6 3 6 6 2 R d 7 c 0 4 1 5 6 3 6 6 2

: a (4 6 1 3 2 5 0 a 0 9 0 0 2 6

: c b (5 6 b c 0 7 6 0 0 0 0 3 9 3 c { ]01[ )b3 1-(: ( 1 )c,b( (2

: : (1 09=6+)3+2+ 0+ 2 + 4 + 3( + )2 + 0+ 0 + 7 + 0 + 1(3 05 { ]01[ 0

. : (2

]01[0{R + )9 + 0 +1 + 6 + 5 +9(+)3 + 2 + 3+ 0+ 0+6(3

R + 02 + )41(3 { ]01[ 0 R + 02 + 24 { ]01[ 0 R + 2 { ]01[ 0

8=R R{801@ > @01>0{R+)2+6+6+1+0+7(+)6+3+5+4+d+c(3(3

+d+c(3 @01>0{R+)22(+)81 @01>0{R+22+45+d3+c3

@01>0{R+6+d3+c3 @01>6- d3- c3-{R

: )I( @01>)2- d- c-(3{R @01>0{R+)2+6+6+1+0+7(+)6+3+5+4+c+d(3

+c+d(3 @01>0{R+)22(+)81

@01>0{R+22+45+c3+d3 @01>0{R+6+c3+d3

@01>6- d3- c3-{R )II( @01>)2- d- c-(3{R

)II( )I( @01>0{6+)6+0+9+a+5+3(+)2+0+0+0+2+1(3(4

@01>0{6+)a+32(+)5(3 @01>0{a+6+32+51

@01>0{4+a @01>4-{a

@01>6{a: 6=a

@01>0{6+)3+3+0+0+7+c(+)9+0+0+6+0+b(3(5

@01>0{3+)31+c(+)51+b(3 @01>0{61+c+54+b3

@01>0{1+c+b3 @01>)b3-1-({c

)c,b( @01>)b3-1-({c

@01>)b7+9({c : @01>7{3- @01>9{1- :

bdd 09: c b 9=c: 0=b

6=c 1=b 3=c 2=b

0=c 3=b 7=c 4=b 4=c 5=b 1=c 6=b 8=c 7=b 5=c 8=b 2=c 9=b

: )c,b( `)2,9(,)5,8(,)8,7(,)1,6(,)4,5(,)7,4(,)0,3(,)3,2(,)6,1(,)9,0(^

II

:1

12 { ]5[ 61

16 { ]7[ 9 441 { ]7[ 11

2)541( { ]21[ 11-

:2 :

7 7253 11 5398 31 1277

Z :3 b { ]9[ 3 a { ]9[ 2 (1

b + a3 { ]9[ 0 5 2b2 + 2a b { ]5[ 3 a { ]5[ 2 (2

:4 7002 80026002

:5 0001 { ]73[ 1 (1 n301 { ]73[ 1: n (2 73 7301001 (3 :6

b a : 5 = b , 72 = a

71 = b , 684 = a 5 = b , 83- = a

8- = b , 621- = a

:7 8 21 a

3 6 4 a

" " :8 ( )

x . B 2 A 1 1 62 x x )01,1( -

01 + x = y 62 x 1 y . (1 DOBMOC (2 TAEROLACAB (3

:1 5 5 5=61-12 *

12 { ]5[ 61 7 25 25=9-16 *

16 { ]7[ 9 7 331 331=11-441 * 441 { ]7[ 11 2)541( { ]21[ 2)1( 541 { ]21[ 1 *

2)541( { ]21[ 1 1 { ]21[ 11-

541 { ]21[11- : 2

7 725 3 x

7253 { ]7[ 0 53 { ]7[ 0 .11 5398 x

5398 { ]11[ 531 98 { ]11[ 1 5398 { ]11[ 1

31 1277: x 77 { ]31[ 1- 77 { ]31[ 21

1277 { ]31[ 21 1277 { ]31[ 1- 12 1277 { ]31[ 2)1-( :30

a3 { ]9[ 6 a { ]9[ 2( 1 b { ]9[ 3

b + a3 { ]9[ 0 b a3+ { ]9[ 6+3 (1 .....) 2a { ]5[ 4 a { ]5[ 2 ( 2

2b { ]5[ 4 2b { ]5[ 9 b { ]5[ 3 ( 2 ....) 2b2 { ]5[ 3 2b2 { ]5[ 8

2b2 { 2a+ ]5[ 4+3 ( 2) ( 1) 2b2 { 2a+ ]5[ 2

2 5 2b2 2a+ : 5

999 u 9997372 999=1-0001 (1 0002 2 73 7301001

:6 b a

72 = a , 5 = b (1 72 = 5( 5 + )2

2 = r , 5 = q 8 - = b , 621 - = a (2 +621 = 8( 51 + )6

6 = r , 51 = q 5 = a , 83 - = b (3 83 = 5( 7 + )3 83 = 5( -7 ) 3 83 = 5(-7 ) 5 + 5 3 83 = 5(-8 + )2

2 = r , 6- = q :7

a { ]21[ 8 8 21 a 8 4 a a{u 843@ >

8 3 a 8 6 a a{u 862@ >