**************************************

...202 00 144

**************************************

( ;14PGCD 40)2020 ...

BAC2020

larbibelabidi@gmail.com

Facebookالجزائري العربي

1،،2

BAC2020

10291010 2

1)75

2 ,a b75ab

;x y( 1)( 1) 75x y

2 ;x y

12 2 15x y 22 24 3x y 33 2xy x y 425 49xy y

n

1134n 22n 25 7n 1235 6n 8n

1d1441276

21441276

3

uv1440 276u v d

n21685n37

33519n53n

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172

, 9

a b

PGCD a b2

360

, 6

ab

PGCD a b

3

2 2 600

, 5

a b

PGCD a b3

2 22 2112

, 8

a b

PGCD a b

n 2 5a n n

1a2a3

a6

02

05

06

07

01

03

04

10291010 1

52k14k

2 52kk

1 417 k5

3 4 3 4 22 17 3k k 5 1999 2008 492007 61 87 5

n126 n7

2n5n7

3n23 6 5n n

nA7

1n7n9

2n7 3 1n n9

1n2 2 13 2n n7

2n2 1 22 3 1n n11

3n2 1n n

x2x52 25 3x y xy

1n32 2n n7

2) n2 3 6 0 11n n

3n2 3 6n n121

1n2n7n

08

09

20

22

21

23

24

10291010 3

9 313548 2537

2n4 3 25 5 5 5 0 13n n n n

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123456n

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251 13 35

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7

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x (1 y

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413a (3)

102b

ab

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26

29

27

28

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10291010 4

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2 ,x y 1 . . . 2045 64 1x y

1 2045,64PGCD 12

2 12 1

S,x y

11 3 65............. 1x y

1,x yS2

0 02 3 11x y21

2,x yS5x5y

x, y

1 , 60

180

ppcm x y

xy

x y2

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105

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x y

x y

3

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3 ,

405

ppcm x y xy

x y

4

( , ) 100

gcd( , ) 5

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p x y

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1n26 17 12n n

2n22 4 1a n n 2b n

ab

12

11

13

14

15

10291010 5

1 (x;y)5x 3y 1...(E) xy

(6n 2;10n 3) (E)n

10n 36n 2

2a 10n 3 b 3n 5 dab

d 1d 41

d 41 n 12 41

3A 20n² 36n 9 B 6n² 19n 15

AB2n 3

nnAB

1k 5k4 1 11

2nn411

3n5n 3 10n(2 2017 3 1438 1) 11

4n5n 2(2 2017 n 3) 11

1nn35

2201714375

3n4n 3 2n 1(48 2 9 1) 5

4n4n n(3 27 4) 5

(E) (x;y)6x 7y 19 xy

10 0(x ;y )(E)0 0x y(E)

2

24 7

5 6

42

3(x;y)(E)x y 1 13

171027

181027

1910262

161029

10291010 6

4n57

n

nn 5 2020 7

n 1437 6

n ،n813العدد الطبيعي أ( -1

:2015 200742 138 2014 3 13

n ( أ-2 n 2n 3 2n(5n 1) 64 5 (5n 6)8 13

(n n 2n 3(5n 1) 64 5 0 13

xy(E)(x;y)11x 7y 1

10 0

(x ;y )(E)0 0

x y 1

(E)

2abSS 11a 1

S 7b 2

(a; b)(E)

S77

1nn911

22012201111

3n15n 1 10n 20124 9 4 2011 2011 11

4n20122011 2n 2 11

(S)

x 3 15

x 6 7

x

1153(S)

3010252

3210232

3110212

3310211

10291010 7

20

x(S)x(S)

0

0

x x 0 15

x x 0 7

3(S)

4153

76

511611

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nA 2 3 4 5 6

1 4 3 7 3A 6 7

2nn2n37

3nn

A 17

2011A7

41432

A7

n7

n 11 00

1n3

2n5

n15

34 n

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2 2008 2008(10 )² 10 1 0 13

3n 2n n10 10 1 0 13

1y' (ln2)y

2f f (0) 1f (x)

3410221

3510202

3610201

3710091

10291010 8

3n7n2

7f (2009) 4

4nn

S f(0) f (1) ...f (n)

nn

S7

n5

1aba n 2 b 2n 3

ab

ab7n 57

nPGCD(a;b) 7

2pqp 2n² 7n 15 q n² 7n 10

pqn 5

nnPGCD(p;q)

xy 4x 9y 319....(1)

1(82;1) 11

2(a;b)4a² 9b² 319....(2)

0 0(x ;y )1

0x

0y

3810082

3910081

10291010 9

14035

1

2

2(x;y)1009x 2017y 1 .

3a

a 2019 2017

a 2019 1009

4nnn79

L7

ة2018 مر

L 111........1

42L9

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11421(E)

(x;y)(E) x 3 5

(E)21 0141 01

6

321171119(a;b)

2m d 2017 d PGCD(a;b)m PPCM(a;b)

nu0u 1

n 1 nu 7u 8: n

1nn 1

n3u 7 4

2nn

nS 1 7 7²...... 7 n 0 1 nS ' u u ...... u

nnSnSnS'

nn 2

n18 S' 7 24n 31

3nn75

nnS'5

4210272

4110271

401028

10291010 20

(E)xy63x 5y 159...(E)

1563(E)

2(x;y)(E) x 3 5(E)

35 0 710 0 5

2

4nn35

nx y 20173 4n 1438 5

(x;y)(E)x

n(u )0u0u 0

n 1 nu 4u 1

1nn

n

1u (4 1)

3

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2n(v )nn n

1v u

3

n(v )0v

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3nn4 1n 14 1

4nn47

nnA6n 4

n nA 9S 6n 3 7

n(u )0uq

1 2

4 3

1 2

ln(u ) ln(u ) 11

u u e (1 e )

1u1u2q

24

1u e3q eunn

n 0 1 nS ln(u ) ln(u ) ..... ln(u ) Snn

3nna n 3

3n n n

PGCD(2S ;a ) PGCD(a ;14)

431027

441027

4510262

10291010 22

n nPGCD(2S ;a )

n nPGCD(2S ;a ) 7

42n

7

52016

n n nb 3n.a 2S 1437 1

n

n

b 0 7

n 0 5

6n9n 1 12n 11437 3 4 52 7

13n

7n

11

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2(E) (x;y)7x 3y 8 xy

(E)

dxy(x;y)(E)

d

(x;y)(E)d 4

(x;y)(E) 7x 3y2016 1437 0 11

n ،n27يعي العدد الطب أ( -1

:1954 1962 531962 1954 2015 7

. 897832( أ-2

981977

3xy2

xy

x² y² 31328

x y 8 22

4 (abcabac

ab c

nnPGCD(a;b ) 1PGCD

1954196219621954

4610261

4710252

10291010 21

12013x 1962y 54: (E) xy

PGCD(2013;1962)(E)

(x;y)(E) x 0 6

0 0(x ;y )

074 x 80(E)

2dxy(x;y)(E)

d

ab671a 654b 18 PGCD(a;b) 18

.1n32n 14n 2 n 3

PGCD( ; ) PGCD( ;10)

PGCD( ; )

nPGCD( ; ) 5

1n411

n

5n n4 4 n 0 11

n 2 10

2n 2n 27 0 n 1

(a;b)(b a)(a b) 24

24

1

10141 3403

4810242

4910232

5010233

10291010 23

(a;b)²b² a² 24

a b 9

321131434

671478

²(x;y)2013x 1434y 27

²(x;y)2011x 1432y 31....(1)

12111

0 0(x ;y )11

2nn27

2012143220117

n n n n2010 2011 1432 0 7

3N29

( ; ) 1

N

113x 7y 1...(E) x y(E)

2a

a 1 7

a 0 13

3nn9713

4b900 086

0

b91

5210212

5110222

10291010 24

1n3n3 113

2 n

3n 13 3 3n 23 9 13

3 n n3132010200513

4 pp 2p 3p

pA 3 3 3

p 3np

A13

p 3n 1 p

A13

pA13p 3n 2

5ab3

a 1001001000b 1000100010000

abp

A

ab13

1²7x 65y 2009....(1)

(x;y)1y7

1

2nn29

3n6n2 3n 2 9

4n6n

nu 2 1

nu9

1 2(7u )x (u )y 126567....(2) (x;y)xy

0 0(x ;y )2

0x

0y

0y 25

5310202

5410201

10291010 25

x1y

AxA 5566

1(5x² 6)(x 1) xy

A (5x² 6)(2 2y)

xyx12

A

2584

aba ba b 32

a² b² 584

xy3x 21y 78....(E)

1(E)²

(x;y)²(E) x 5 7

(E)

2nn57

(x;y)²(E) x y5 5 3 7

5510092

5610081

10291010 26

10291010 27

12491327852915

2²7x 6y 79 79772

32915

249132785

1993171

1²1xy993x 170y 143....(1)

0 0(x ;y )1

0 0x y 6

²1

3A(A 1)1986341

14311

1x 3x 5 0 11

2²3x 11y 5....(1)

3dxy

d(x;y)1

4(x;y)²1d 5

²13x 11y 23....(1)

10 0

(x ;y )10 0

x y 1 1

(x;y)110 x 40

2xydd

(x;y)123 d

(x;y)x

572997

581002

592998

601008

10291010 28

1995

(x;y)²x 7y 1995 PGCD(x;y) 19

abc1 a b c

abcabcab c 46 bc 555

1199614972994

21996x 1497y 3994....(1) xy

x3 y21

(x;y)1x.y 1950

11885581

21885x 580y ..(1)

1²

31305 1(x;y)xy

1²18x 4y 84

(x;y)x.y 0

2n305557

n

²5x 6y 3....(1)

1(x;y)1x3(1)

21

x 1 6

x 4 5

3(x;y)²1

(x;y)(x² y²) 56

622995

612996

632997

642991

652991

662990

10291010 29

1xy(x y)xy

2215 229 30

3xy

(x;y)(x y)2292 215(x y )

1²(x ';y ')9 ' 14 ' 13 x y 3,1

2² 45x 28y 130

(x;y) x2 y5

3N2 395 67

N

n² n n 2 n

1PGCD( ; ) PGCD( ;n)

PGCD( ; )

2abn a 3520b 384

3n 2 ab

nPGCD(a;b) 3n 2 PGCD(a;b) 2(3n 2)

PGCD(a;b) 41

1n2n11

nn 2a 2n 1b 4n 3c 2n 3

1ababc

2n bc

n PGCD b,c 3PPCM b,c 1305

32ba

672996

682997

691005

701007

722994

10291010 10

14141994

2n n

u2n

nu

n nu

n1 25 2 5 2 ....... 5 2n

nS

nnS11

1n7n11

k 4 4 1 4 2 4 37 7 7 7 k k k k11

221 7 7 ... 7 n

nS

n 4 10 n nS S

nnS11

11981

2ab2 25 1980m d

,m PPCM a b,d PGCD a b

4862 1430 2002.......( ) x y Exy

1486214312112

2E2

E

3ab ,a bE,d PGCD a bd

d

,a b7d

n124 n5

23n5

3200914295

422 4 3 n n

nA

nnA5

711004

732980

741007

751009

10291010 12

1n2n 3 3n 9n 16

n33n 11n 48n 3

n23n 9n 16

2abc

PGCD a;b PGCD bc a;b

32

3PGCD 3n 11n;n 3 PGCD 48;n 3

448

n33n 11n

An 3

a A 0 g f a a g f a a

a aba 11n 3b 13n 1

1 ab51

1 E109x 226y 1 xy

109226 E

E

141 226 ;68 109 k kk

d226

e109d 1 226e de

2227

3A aa 226fgA

fa109a227g a141a227

g f 0 0

a A 0 226a 1 2271

761003

771005

782982

10291010 11

2 250x 11y 3

3 n51 ab

n25ab

1²3x 8y 5........(E) E

8k 1;3k 1 k

2nxyn 3x 2

n 8y 7

x,y E

S

n 2 3

n 7 8

n

n S n 23 24

3k2k2327 k8

1991 S20081991 124

1) Exy11x 5y 2

(2;4) E

(x;y) E11(x 2) 5(y 4)

E

2n a 5n 2 b 7n 5

5b 7aPGCD a;b 1PGCD a;b 11

1n PGCD a;b 11

1 2010

111............11م رق مرةال

N

1N11

22111201010 1 9N

21119N

2111N

3N22121

791008

801008

821022

10291010 13

0,1,2,3,4,5,6,7,8,9, , 12

11N1212

1 1N

1N

2N2 1131N

2N12

2N12

1 0...... nN a a a

3 0N a 3312

122N12

4 1 0... 11 nN a a a11

12

121N11

4N1212

4x y

xyN33

1ab ;PGCD a b ab p p

p2a2 ( )a a a b ab

papb

;PGCD a b p

2aba b

PGCD a;b 5

PPCM a;b 170

PGCD a b;ab 5

PPCM a;b 170

I E11 7 5x y xy

;u v11 7 5u v

;u v E E

811008

382010Antilles Guyane

482210Antilles Guyane

10291010 14

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11 7 5 0x y E ;M x y

0 50x 0 50y

DE

II F2 211 7 5x y xy

;x y F 2 22 5x y

xy

2x22y5

;x y Fxy5

IIIxy5 x;y F

F

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pp1 1pa

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1 1u2u3u

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nu

nnu

32u

nu

4nnu235

5n 3 4 1 11n

nu

nnu11

6 1610 1 17

k16 8ku 17

543210x

52x

543210y

522y

851022 Polynésie

10291010 15

1251

22118

2118

ab 3 3m 35d 2008

m PPCM(a;b)d PGCD(a;b)

1nn57

2n2 n

nS 1 5 5 .... 5 n 1

n4S 5 1

a n4S a 7 nS 2a 7

2016S7

3 n

0 nE :5 x S y 0 n

nE :5 x S y 7

nnSn5 0E

E x;ynx 35 kS

ny 28 k5 k

25x 31y 7

PGCD(x, y) 7

1nn411

10n 4 312015 10 1434 11

2x x² 2x 9 0 11

n 4n 1 2n 15 9 2 2 11

3n 4n 1 2n17 3 9 5

554n 4n 1 5n 1187 17 11 3 60 4

nab8 1 a n7 1 b n

861008

87

88

89

10291010 16

1ab

223 26 1 x y( )Exy

n( ; )a b( )E

( )E( )E

3a

0 25

23 1 26

a

a

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0 1u n1 10 81n nu u

21u2u3u

1n110 9n

nu

n28 9 9 10 9 10 ... 9 10n

nu

nnu

32u

nu235

4nnu7

5n 2 ( 1) 11nnu

nnu11

23x 2y 1 ( )E

n(14n 3;21n 4) ( )E

14n 321n 4

d2n 121n 4

d 1d 13 n 6 13d 13

nn 23 2B 28n 8n 17n 3 2A 21n 17n 4

AB( 1)n

nAB

90

92

10291010 17

11885581

21885 580 ...... 1x y

1

31305

1

,x yxy

2( ).....32x 7y 8E 1 5

1)( );x y ( )E4 3 7x

2 )2( )E

3)n2 8957

n

42017

a2( 2017)a

1( ; )a b2;( ) 48PG a bCD( ) 2160;a bPPCM

2x9 17 5x

3432 240 816x yxy

4n9253

n4517

n19802025

520162016

20181202x

23x 2y 1 ( )E

n(14n 3;21n 4) ( )E

14n 321n 4

91

93

94

95

10291010 18

d2n 121n 4

d 1d 13 n 6 13d 13

nn 23 2B 28n 8n 17n 3 2A 21n 17n 4

AB(n 1)

nAB

1n3n5 2018(1439)1954(1962)5

2n

4 3 1954 2018(2018) 2 (1962) (1439)n5

3n4 13 2017 6n n5

1n3n

2n 16 2 8 133 2 109 11 0 10 n n

3n 17 3 1 0 10 n10 25 n

4Axx0xx023y6127

xyA

A

N( )S5 13

:1 17

N

N

239( )S

N1 17 5 13N x yxy

17 13 4x y

2Z17 13 4x yxy

k18 221N k( )S

96

97

98

**************************************

********************

********************

BAC2020

larbibelabidi@gmail.com

Facebookالجزائري العربي

4242 53

3-(6n 2;10n 3) (E)

(6n 2;10n 3) (E)5(6n 2) 3(10n 3) 1.

10n 36n 2

5(6n 2) 3(10n 3) 1 10n 36n 2

4d 1d 41

PGCD(a;b) da 10n 3 b 3n 5

abn

3a 10b 10(3n 5) 3(10n 3) 41 3a 10b 41

dada31

dbd10b2

12d3a 10bd41

41141

d 41 n 12 41

d 41

a 0 41

b 0 41

10n 3 0 41

3n 5 0 41

7n 2 0 41 7n 2 41 42n 12 416

n 12 41 42 1 41

5AB2n 3

A 20n² 36n 9 B 6n² 19n 15

A 20n² 36n 9 (2n 3)(10n 3) A2n+3

B 6n² 19n 15 (2n 3)(3n 5) B2n+3

nnAB

PGCD(A;B) (2n 3)PGCD(a;b)

d 1d 41

d 41PGCD(A;B) 41(2n 3) n 12 41

d 1PGCD(A;B) (2n 3) n 41k 12 k

48423;

4242 54

3k 5k4 1 11

54 1024 11.93 1 54 1 11

k 5k4 1 11

4n433n411

n4115

5k 45k 35k 25k 15kn39541n4

5n5n 3 10n2 2017 3 1438 1 11

10n 10n1438 (8) 11 1 11 5n 3 5n 32017 4 11 9 11

5n 3 10n2 2017 3 1438 1 2 9 3 1 1 11

5n 3 10n2 2017 3 1438 1 22 11 5n 3 10n2 2017 3 1438 1 0 11

6n5n 22 2017 n 3 33

5n 22 2017 n 3 11 5n 22 2017 n 3 0 11

5n 2 5n 22017 4 11 5 11

5n 22 2017 n 3 0 11 n 7 0 11 n 11p 4 p

3nn35

03 1 5 13 3 5 23 4 5 33 2 5 43 1 5n354

4k 34k 24k 14kn 2431n3

2201714375

1437 2 5 1437 ( 3) 5 2017 20171437 ( 3) 5

2017 2017 4(504) 1( 3) ( 1) .3 5 20171437 ( 3) 5

4(504) 13 3 5 2017( 1) 1 2117

20171437 ( 3) 5 20171437 2 5 ( 3) 2 5 2017143752

494239

4:4239

4242 55

5n4n 3 2n 1(48 2 9 1) 7

4n 3 2n 1(48 2 9 1) 5 4n 3 2n 1(48 2 9 1) 0 5

48 3 5 4n 3 4n 348 3 5 4n 348 2 5

2n 1 4n 29 3 4 5

4n 3 2n 1(48 2 9 1) 2 2 4+1 5 4n 3 2n 1(48 2 9 1) 5 5

4n 3 2n 1(48 2 9 1) 0 5 ( 5) 0 5

4n4n n(3 27 4) 54n n(3 27 4) 5 4n n(3 27 4) 0 5

3n27 (3) 5 4n3 1 5

4n n(3 27 4) 0 5 3n(3 ) 3 5

3n 1 4 n 3 4n 4p 3 p

10 0

(x ;y )(E)(E)

0 0(x ;y )(E)

0 0x y

0 06x 7x 19

0 0x y 19

0 0

6x 7y 19

6x 7x 19

0 06(x x ) 7(y y ) 0

0 06(x x ) 7(y y ).....(1)

160

7(y y )60

(y y )67

0(y y ) 6k y 6k 19

y1x 6k 19

(E)(x;y) (7k 19;6k 19) k

4

24 7

5 6

42

24 7

5 6

7 24

6 5

6 7 19...(E')

(E)(E ')x 7k 19

6(7k 19) 5 42k 119

42k 119 42k 7 119 7 42

427

5(x;y)(E)x y 1 13

4;42383

4242 56

(x;y) (7k 19;6k 19) x y 1 13

7k 6k 39 13 13k 39 13 k 3 1 2 k 4

k 2;3;4(x;y)( 5; 7) (2; 1)(9;5)

6n59

05 1 7 15 5 7 25 4 7 35 6 7 45 2 7 55 3 7 65 1 7n576

6k 56k 46k 36k 26k 16kn 326451n5

n

nn 5 2020 7

n 1437 6

nn 5 2020 7

n 1437 6

nn 5 4 7

n 3 6

6k 36k 3 5 4 7

n 6k 3

k 0 7

n 6k 3

k 7p

n 6k 3

n 42p 3 p

3(n835

08 1 13 18 8 13 28 12 13 38 5 13 48 1 13n8134

4k 34k 24k 14kn 51281n8

2015 203742 138 2014 3 35

42 3 13 138 8 13 2015 2015138 8 13

2015 3 4 20158 5 13 2015138 5 13

2014 12 13 2014 1 13 12 1 13

2037 20372014 ( 1) 13 20372014 ( 1) 13 2037

2015 203742 138 2014 3 3 5 1 3 13

2015 203742 138 2014 3 11 13 2015 203742 138 2014 3 1311

4(n n 2n 3 2n(5n 1) 64 5 (5n 6) 8 13 n 2n64 8 5 8 13 2n 3 2n 35 ( 8) 13

5242373

4242 57

2n 3 2n 2n 3 3( 8) (8) ( 1) .8 13 2n 3 2n( 8) 5.(8) 13

n 2n 3 2n 2n(5n 1) 64 5 (5n 1) 8 5 8 13

n 2n 3 2n(5n 1) 64 5 (5n 6) 8 13

n n 2n 3(5n 1) 64 5 0 13

n 2n 3 2n(5n 1) 64 5 (5n 6) 8 13

n 2n 3(5n 1) 64 5 0 13 2n(5n 6) 8 0 13

(5n 6) 0 13 8132 n813

(5n 6) 0 13 5n 6 13 5n 20 13 6 20 13

5n 20 13 n 4 13n 13p 4 p

10 0

(x ;y )(E)0 0

x y 1

0 0(x ;y )

0 0

0 0

11x 7y 1

x y 1

0 0

0 0

11x 7y 1

7x 7y 7

04x 8

0x 2

0 0y 1 x 3

0 0(x ;y ) (2; 3)

(E)

0 0(x ;y ) (2; 3) (E)

11x 7y 1

11(2) 7( 3) 1

11(x 2) 7(y 3) 0

11(x 2) 7(y 3) 0 11(x 2) 7( y 3)......(*)

(*)7 /11(x 2)7 / (x 2)711

x 2 7k x 7k 2 k

x(*)11(7k) 7( y 3) y 11k 3

(E)(x;y) (7k 2; 11k 3)

2(a; b)(E)

S 11a 1

S 7b 2

11a 1 7b 2 11a 7( b) 1

11a 7( b) 1 (a; b)(E)

S77

5342353

4242 58

(a; b)(E)(a; b) (7k 2; 11k 3)

a 7k 2 aS 11a 1 S 11(7k 2) 1 77k 23

S 77k 23 23S77

1nn911

09 1 11 19 9 11 29 4 11 39 3 11 49 5 11 59 1 11n9115

5k 45k 35k 25k 15kn

1153491n9

22012201111

2011 9 11 2012 20122011 9 11

2012 2 5 20122011 4 11 2012 5k 29 9 4 11 20122011114

3n15n 1 10n 20124 9 4 2011 2011 11

15n 1 5n 39 (9 ) 9 9 11 5n(9 ) 1 11

10n 5n2011 (9 )² 11 10n2011 1 11 5n(9 ) 1 11

20122011 4 11

15n 1 10n 20124 9 4 2011 2011 4 9 4 1 4 11

15n 1 10n 20124 9 4 2011 2011 0 11 4 9 4 1 4 44 4 11

4n20122011 2n 2 11 20122011 2n 2 11 20122011 2n 2 0 11

20122011 2n 2 0 11 4 2n 2 0 11 20122011 4 11

2n 6 11 n 3 11

n 8 11n 11 8

1153(S)

153 15 10 3

153 7 21 6

153 3 15

153 6 7

153(S)

2x(S)

0

0

x x 0 15

x x 0 7

5442343

5542344

4242 59

10

x(S) x(S)

0

0

x x 0 15

x x 0 7

0

0

x 3 15....(1)

x 6 7

x(S)

x 3 15....(2)

x 6 7

12

0

0

x x 15

x x 7

0

0

x x 0 15

x x 0 7

2

0

0

x x 0 15

x x 0 7

0x(S)x(S)

0

0

x x 0 15....(2)

x x 0 7

0

0

x 3 15....(1)

x 6 7

12

x 3 15

x 6 7

x(S)

3(S)

x 3 15

x 6 7

7x 21 105 .....(1)

15x 90 105 ......(2)

x 153 1051

x 48 105 153 48 105x 105k 48 k

4

x

153x 15 3

76x 7 6

x(S)500 x 600

3x 105k 48 500 105k 48 600

500 105k 48 600452 105k 552452 552

k105 105

k 5

kx 105k 48 x 573

1 4 3 7 3A 6 7

7 0 7 4 3 0 7 4 3 7 n n n n n

nA 2 3 4 5 6 3 3 3 3 3

3A 2 3 4 5 6

5642334

4242 5:

4 3 7 5 2 7 6 1 7

3 3 3 3 3

3A 2 3 ( 3) ( 2) ( 1) 7

3 3 3 3 3

3A 2 3 (3) (2) (1) 7

3A 1 7 3

A 6 7 1 6 7

2nn2n37n27

02 1 7 12 2 7 22 4 7 32 1 7n273

3k 23k 13kn 421n2

n37

03 1 7 13 3 7 23 2 7 33 6 7 43 4 7 53 5 7 63 1 7n3116

6k ' 56k ' 46k ' 36k ' 26k ' 16k 'n 546231n3

3nn

A 17

n n n n n

nA 2 3 4 5 6 nn 2p 1

2p 1 2p 1( a) a 2p 1 2p 1 2p 1 2p 1 2p 1

nA 2 3 3 2 1 1 n

A 1 0 7

2011A7

2111 2011A 1 0 7 2011

A 6 76

41432

A7

1432 1432 1432 1432 1432

1432A 2 3 4 5 6

2p 2p( a) a 1432 1432 1432 1432 1432 2864 1432

1432A 2 3 3 2 1 2 2.3 1

2864 3(954) 2 3k 21432 6(238) 4 6k ' 4

2864 1432

1432A 2 2.3 1 4 2.4 1 7 1432

A 6 76

4242 5;

1n3

n7

n 11 00 0 6

4 3 0n 11 00 1.7 1.7 .7² 0.7 0.7 49 2744

n3 0 3

0 3 49 2744 0 3 2 0 3 49 1 3 2744 2 3

2 0 3 1 3 3k 1 1 4 0 6

2n5

n5 0 5

0 5 49 2744 0 5 1 0 5 49 1 5 2744 1 5

1 0 5 4 5 5k 4 4 0 6

n15

n15n5 3

4 12

34 n

n 49 2744 n 49(4) 2744 2940

1nn1013

010 1 13 110 10 13 210 9 13 310 12 13 410 3 13 510 4 13 610 1 13n10136

6k ' 56k ' 46k ' 36k ' 26k ' 16k 'n 43129111n10

2 2008 2008(10 )² 10 1 0 13

2008 6(334) 4 6k ' 4 200810 3 13

2008 2008(10 )² 10 1 3² 3 1 13 2008 2008(10 )² 10 1 13 13

2008 2008(10 )² 10 1 0 13

3n 2n n10 10 1 0 13

5742323

5842324

4242 62

n

6k ' 56k ' 46k ' 36k ' 26k ' 16k 'n 43129111n10

3913912n10

7111732n n10 10 1

n 6k ' 2 n 6k ' 4 k '

1y' (ln2)y

y' ay b ax by c.e

a

y' (ln2)yxln2 xy c.e c.2 c

2 f (0) 1f (x)xy f (x) c.2 f (0) 10c.2 1c 1xf (x) 2

37n2

02 1 7 12 2 7 22 4 7 32 1 7n273

3k 23k 13kn 421n2

7f (2009) 42009f (2009) 22009 3(669) 2 3k 2

3k 2f (2009) 4 2 4 7 f (2009) 4 0 7 3k 22 4 7

7f (2009) 41

4nn

S0 2 n

nS f (0) f (1) ...f (n) 2 2 2 ....2

nS12

n 1

n 1

n

2 1S 1. 2 1

2 1

n

nS7

nS7 n

S 0 7

nS 0 7 n 12 1 7 n 1 3k n 3k 1 k *

59422;4

4242 63

ab

d PGCD(a;b)b 2a 7

dabdb 2ad7d 1d 7

ab7n 57

ab7

a 0 7

b 0 7

2n 3 0 7 ....(1)

n 2 0 7 .....(2)

21 n 5 0 7 n 57

nPGCD(a;b) 7

PGCD(a;b) 7

2n 3 0 7 ....(1)

n 2 0 7 .....(2)

n 5 0 7

n 5 0 7 n 5 7k n 7k 2 k1

pqn 5

p 2n² 7n 15 (n 5)(2n 3) pn 5

q n² 7n 10 (n 5)(n 2) pn 5

nnPGCD(p;q)

p 2n² 7n 15 (n 5)b q n² 7n 10 (n 5)a

PGCD(p;q) (n 5)PGCD(a;b)

PGCD(a;b) 1PGCD(a;b) 7

1PGCD(a;b) 1PGCD(p;q) (n 5) n5n 7k 2

2PGCD(a;b) 7PGCD(p;q) 7(n 5) n5n 7k 2

5:422:3

4242 64

1(82;1) 11

(82;1) 14(82) 9(1) 319

4x 9y 319...(1)

4(82) 9(1) 319.....(2)

214(x 82) 9(y 1) 0

4(x 82) 9(y 1).....(*)

(*)94(x 82)9(x 82)94

(x 82) 9k x 9k 82 x(*)y 4k 1

(x;y)(x;y) (9k 82;4k 1) k

2(a;b)4a² 9b² 319....(2)

4a² 9b² 319....(2) (2a 3b)(2a 3b) 319

319

319 11 29 (1 1)(1 1) 4 319D 1;11;29;319

1(a;b)2( a; b)

22a 3b 2a 3b

1(2a 3b)(2a 3b) 1 319 2a 3b 1...(1)

2a 3b 319.....(2)

21

4a 320a 80a1b 53

2(2a 3b)(2a 3b) 11 29 2a 3b 11...(1)

2a 3b 29.....(2)

21

4a 40a 10a1b 3(a;b)

(80;53)( 80; 53) ( 80;53)(80; 53)(10;3)( 10; 3) ( 10;3)(10; 3)

0 0(x ;y )1

0x

0y

4a² 9b² 319....(2) 0 0

4x 9y 319....(2) 0

x a²0

y b²

0 0(x ;y )1

0x

0y

(640;2809)(100;9)

5;422:4

4242 65

32

4035

1

4035

1

2 4036

1

2018

1 2017

2

0 0x y 1

2

0 0(x ;y ) ²

0 01009x 2017y 1 0 0(x ;y ) (2; 1) 2

PGCD( ; ) 12

4(x;y)1009x 2017y 1 .

1009(2) 2017( 1) 1 1009(2) 2017(1) 1

1009(2) 2017(1) 1 (2; 1)1009x 2017y 1...(1)

1009x 2017y 1

1009(2) 2017(1) 1

1009(x 2) 2017(y 1) 0 1009(x 2) 2017(y 1).....(1)

111192017(y 1)1119(y 1)21171119

(y 1) 1009k y 1009k 1

y1x 2017k 2

(x;y) (2017k 2;1009k 1) k

3a

a 2019 2017

a 2019 1009

a 2019 2017

a 2019 1009

a 2 2017

a 1 1009

2019 2 2017 2019 1 1009

a 2017 2

a 1009 1

1009 2017 1...(2)

12y 1009k 1 x 2017k 2

62423:

4242 66

a 2017(1009k 1) 2 2035153k 2019 k

6n7;

07 1 9 17 7 9 27 4 9 37 1 9n793

3k 23k 13kn 471n7

42L;

L7 L 111...1

0 1 2017L 111...1 7 7 7² .... 7

L1720187 1

L6

20187 1L

6

201942L 7 7 42

20197 7 0 9 20197 7 9

311421(E)

11421

1104 20 5 4 20 4 5 0

114214

23104 2 13 20 2² 5

114212² 4

104x 20y 272....(E)

(E)11421272272 4 68

(x;y)(E) x 3 5

(E)26x 5y 68....(E') (E)4

(E')26x 68 5y 26x 68 5

x 3 5 26 1 5 68 3 5

(E)

(E)(E')

x 3 5x 5k 3 k

6342393

4242 67

(E')5y 68 26x x

5y 68 26(5k 3) 5(21k 2) y 21k 2

(E)(x;y) (5k 3;26k 2) k

2

1 0145 4 3 21 4 4 4 4 1 1025 320 16 ....(1)

1 0164 3 21 6 6 6 1 1267 216 36 ....(2)

121025 320 16 1297 216 36

104 20 272 0 3 0 3

104 20 272 104x 20y 272....(E)

x 5k 3 y 21k 2

0 3 0 3 k 0( ; ) (3;2)

1025 320 16 ( ; ) (3;2) 2017

321171119

2017 44,911009 31,76

211743

111931

(a;b)2m d 2017

d PGCD(a;b)m PPCM(a;b)

d PGCD(a;b)a a '.db b'.dPGCD(a ';b') 1

m d=a×bm=d×a'×b'

m=d×a'×b'2m d 2017 2017

2a ' b' 1 ...(*)d

(*)d2117d 1d 20172117

1d 1(*)a ' b' 1009

(a ';b') (1009;1)(a ';b') (1;1009)1119

2d 2017(*)a ' b' 1

a 'b '

1(a;b) (1009;1)(a;b) (1;1009)

4242 68

3nn 1

n3u 7 4

nu0u 1 n 1 nu 7u 8: n

P(0)

n=003u 30u 1

P(n)n 1

n3u 7 4 P(n+1)n 2

n 13u 7 4

n 1 nu 7u 8: n n 1 n3u 7(3u ) 24

n 1

n3u 7 4 n 1

n 13u 7(7 4) 24

n 1

n 13u 7.7 28 24

n 2

n 13u 7 4

n 1

n3u 7 4 n

2nnSn

nS 1 7 7²...... 7 17

n 1 n 1

n

7 1 7 1S 1

7 1 6

nSnS'

n 0 1 nS ' u u ...... u n 1

n3u 7 4

n 0 1 n3S ' 3u 3u ...... 3u

n 1 n

n3S ' (7 4) (7² 4) ...... (7 4) 7(1 7 7² ....7 ) 4(n 1)

n n3S ' 7S 4(n 1)

nn 2

n18 S' 7 24n 31

n n3S ' 7S 4(n 1) 6

n n18S ' 7(6S ) 24(n 1) n 1

n

7 1S

6

n 1

n

7 118S ' 7(6 ) 24(n 1)

6

n 2

n18 S' 7 24n 31

3nn75

07 1 5 17 2 5 27 4 5 37 3 5 47 1 5n754

4k 34k 24k 14kn 3421n7

6442394

4242 69

nnS'5

nS'5 nS' 0 5 n18S' 0 518

n 27 24n 31 0 5 n 27 n 1 0 5

1n 4k n 27 n 1 0 5 k 3 5n 20k' 12 (k ' )

2n 4k 1 n 27 n 1 0 5 k 3 5n 20k ' 13 (k ' )

3n 4k 2 n 27 n 1 0 5 k 2 5n 20k ' 10 (k ' )

4n 4k 3 n 27 n 1 0 5 k 4 5n 20k' 19 (k ' )

3785

563PGCD(63;5) 1

PGCD(63;5) 1

63 5.12 3 5 3.1 2 3 2.1 1

1PGCD(63;5) 1

(E)

ax by c ²PGCD(a;b)c

63x 5y 159...(E)

(E)²PGCD(63;5) 1159

4(x;y)(E) x 3 5

63x 5y 159...(E) 63x 159 5y 63x 159 5

3x 9 5 63 3 5 159 9 5

3x 9 5 x 3 535

(E)

x 3 5x 5k 3 k

63x 5y 159...(E) 5y 159 63x

x 5k 3 5y 159 63x

5y 159 63(5k 3) 5y 30 63 5k y 63k 6

(E)(x;y) (5k 3; 63k 6);k

5

5 0 7

654239

4242 6:

7 35 0 7² 7 .5 50 1715.....(1)

10 0 55 3 410 0 5 1.5 5 630 125.....(2)

12630 125 50 1715 63 5( ) 159....(3)

3(E)( ; ) (x;y) (5k 3; 63k 6)

( ; ) (5k 3;63k 6) 0 6 0 4

0 5k 3 6 0 63k 6 4 k 0( ; ) (3;6)

2

150 1715 2 50 1717 6

2 50(6) 1717 2017

2

6n37n

03 1 5 13 3 5 23 4 5 33 2 5 43 1 5n354

4k 34k 24k 14kn 2431n3

nx y 20173 4n 1438 7

x y 20173 4n 1438 5 x y 20173 4n 1438 0 5

(x;y) (5k 3; 63k 6) x

(x y) (68k 9) x y 68k 9 68k 83 3 3.3 3 5 68k 83 1 5

1428 3 5 2017 20171428 3 5 20171428 3 5 20173 3 5

x y 20173 4n 1438 0 5 3 4n 3 0 5

3 4n 3 0 5 4n 4 5 n 1 5n 5p 1 p

3nn

n

1u (4 1)

3

n(u )0u 0n 1 nu 4u 1

n

n

1u (4 1)

3 n

P(0)

664239

4242 6;

n=00

0

1u (4 1) 0

3 0u 0

P(n)n

n

1u (4 1)

3

P(n+1)n 1

n 1

1u (4 1)

3

n 1 nu 4u 1 n

n

1u (4 1)

3

n

n 1

1u 4 (4 1) 1

3

n n 1 n 1

n 1

4 4 1 1 1u .4 1 .4 (4 1)

3 3 3 3 3

n

n

1u (4 1)

3 n

nnun 1u

ab0 0(x ;y ) ²0 0ax by 1

n 1 nu 4u 1 n 1 nu 4u 1

(1; 4)n 1 nu 4u 1 nun 1u

4 nv0v

nvqnn 1 n

v q.v

n(v )nn n

1v u

3

n n

1v u

3 n 1 n 1 n n n n

1 1 4 1v u 4u 1 4u 4(u ) 4v

3 3 3 3

n(v )q 40 0

1 1v u

3 3

nnS

n 0 1 2 3nS v v v ..... v

3n 13n 13n 1

n 0

4 1q 1 1 1S v 4 1

q 1 3 4 1 9

5n4 1n 14 1

n

n

1u (4 1)

3 n

n3u 4 1 n 1

n 13u 4 1

n 1 n n 1 nPGCD(3u ;3u ) 3PGCD(u ;u ) n 1 nPGCD(4 1;4 1) 3.1 3 n 1 nPGCD(u ;u ) 1 1

4242 72

6nn49

04 1 7 14 4 7 24 2 7 34 1 7n47

3

nnA6n 4

n nA 9S 6n 3

nA7 nA 0 7

nA 0 7 6n 4

n9S 6n 3 0 7

3n 1

n9S 4 1 n9S 3 7 3n 14 4 7

3n 2

6n 4 23 ( 4) 7

2

6n 4 3n 23 (4) 4 7

6n 4

n9S 6n 3 0 7 3 6n 4 0 7 6n 6 7

n 1 767n 7p 1 p

3u1u2q

n(u )

1 2

4 3

1 2

ln(u ) ln(u ) 11

u u e (1 e )

1 2

4 3

1 2

ln(u ) ln(u ) 11

u u e (1 e )

1 2

4 3

1 2

ln(u ).(u ) 11

u u e (1 e )

11

1 2

4 3

1 2

(u ).(u ) e

u u e (1 e )

u1u2x² Sx P 0 4 3S e (1 e ) 11P e

x² Sx P 0 4

1x u e 7

2x ' u e

2 1u q.u32

1

uq e

u

4unn n 1

n 1u u .q 4 3 n 1 3n 1

nu e (e ) ) e

Snn

n 0 1 nS ln(u ) ln(u ) ..... ln(u )

n (n 1) n

n 1 2 2

n 0 1 n 0. 0.S ln(u ).(u )......(u ) ln (u ) .q (n 1)ln((u ).q )

0u e3q e

3n

2

n

(n 1)(3n 2)S (n 1)ln((e).e )

2

5nn n n

PGCD(2S ;a ) PGCD(a ;14)

3k 23k 13kn 241n4

6742383

4242 73

na n 3

n2S (n 1)(3n 2) 3n² 5n 2

n n2S (n 3)(3n 4) 14 a (3n 4) 14

n n2S a (3n 4) 14

nn n n

PGCD(2S ;a ) PGCD(a ;14)

n nPGCD(2S ;a )

n n nPGCD(2S ;a ) PGCD(a ;14)

n nPGCD(2S ;a )1412714

n nPGCD(2S ;a ) 7

n nPGCD(2S ;a ) 7

nPGCD(a ;14) 7PGCD(n 3;14) 7 n 3 7k

PGCD(n 3;14) 7 PGCD(7k;14) 7PGCD(k;2) 1

PGCD(k;2) 1k2kk 2k ' 1

n 3 7k n 7k 3 k 2k ' 1 n 14k' 4 k '

62n

7

02 1 7 12 2 7 22 4 7 32 1 7

n27

3

7n

n

b 0 7

n 0 5

2016 2016

n n nb 3n.a 2S 1437 1 3n² 9 (3n² 5n 2) 1437 1

2016 20161437 2 7 1 7 1437 2 7 2016 1 3

nb 5n 9 7 5n 9 0 7 15n 6 7 n 6 7

n

b 0 7

n 0 5

n 0 7

n 0 5

5n 0 35

7n 0 35

2n 0 35

n 0 35n 35k 'k '

6n9n 1 12n 11437 3 4 52 79n 1 12n 11437 3 4 52 7 9n 1 12n 11437 3 4 52 0 7

9n 1 3n 31437 (2 ) .2 2 7 12n 1 3n 44 (4 ) 4 4 7 52 3 7

9n 1 12n 11437 3 4 52 2 3.4 3 7 9n 1 12n 11437 3 4 52 7 7

9n 1 12n 11437 3 4 52 0 7

3k 23k 13kn 421n2

4242 74

33n

7n

11

03 1 11 13 3 11 23 9 11 33 5 11 43 4 11 53 1 11n3115

5k 45k 35k 25k 15kn45931n3

07 1 11 17 7 11 27 5 11 37 2 11 47 3 11 57 10 11 67 4 11

77 6 11 87 9 11 97 8 11 107 1 11n71111

10k ' 410k ' 310k ' 210k ' 110k 'n32571n7

10k ' 910k ' 810k ' 710k ' 610k ' 5n896411n7

n5n 4 10n 42 2016 1437 11

10n 4 10n 41437 7 11 10n 41437 3 11 5n 4 5n 42016 3 11 4 11

5n 4 10n 42 2016 1437 2.4 3 11 5n 4 10n 42 2016 1437 0 11

27x 3y 8

7x 3y 8 7x 3y 8 7x 8 3 x 2 3x 3k 2 k

x 3k 2 7x 3y 8 y 7k 2 k

7x 3y 8 (x;y) (3k 2;7k 2) k

d

dxy(x;y)7x 3y 8

d / xd / 7x1d / yd / 3y2

12d / 7x 3yd / 88d D 1;2;4;8

d=4

PGCD(3k 2;7k 2) 4 PGCD(k 2;8) 4

k 2 4 PGCD(4 ;8) 4

PGCD( ;2) 1 2m 1 2m

k 8m 6 k(x;y) (24m 20;56m 44)

(x;y)(E) 7x 3y2016 1437 0 11

6842384

4242 75

3y 3y1437 7 11 7 x 7 x2016 3 11

7x 3y2016 1437 0 11 7 x 3y3 7 0 11 (x;y) (3k 2;7k 2)

7 x 3y3 7 0 11 k k3 7 0 11 511

10p 410p 310p 210p 110pk45931k3 32571k7 773112k k3 7

10p 910p 810p 710p 610p 5k45931k3 896411k7 14471k k3 7

n 10p 5

(x;y) (30p 17;70p 37) p

3(n29

02 1 7 12 2 7 22 4 7 32 1 7n27

3

1954 1962 531962 1954 2015 9

1962 2 7 1954 19541962 2 7

1954 1 3 19542 2 13 19541962 2 7

1954 1 7 19621954 1 7

2015 6 7 2015 1 7

53532015 1 7 532015 1 7 53

1954 1962 531962 1954 2015 2 1 1 7

1954 1962 531962 1954 2015 0 7 1954 1962 531962 1954 2015 71

4(:;

89 9,49

3k 23k 13kn 421n2

6942373

4242 76

144289

229389

417589

512789

89

9:54

783237832 2 11 89

7832(3 1)(1 1)(1 1) 16

7832

97989111111011

9798911102111089

19581782221297989189

391635644422783271288832

7832

7832D 1;2;4;8;11;22;44;88;89;178;376;712;979;1958;3916;7832

;:3;99

981977PGCD(977;981) 1

981 977 1 4

977 4 244 1

1

981977

3xy

PGCD(x;y) 2x 2x 'y 2y'PGCD(x ';y') 1

x² y² 31328

x y 8 22

PGCD(x;y) 2

x ' ² y ' ² 7832

x ' y ' 4 11

PGCD(x ';y ') 1

(x ' y ')(x ' y ') 7832

x ' y ' 4 11

PGCD(x ';y ') 1

x ' y' 11k 4 7832

x ' y ' 11k 4 4

x ' y ' 1958

x ' 981

y ' 977

x 2x ' 1962

y 2y ' 1954

6ab c

4242 77

abc0 0

(x ;y )0 0

(x ' ;y ' )

0 0ax by 1...(1)

0 0ax ' cy' 1...(2)

ab c(u;v)au (bc)v 1

120 0 0 0

(ax by )(ax ' cy' ) 1

0 0 0 0 0 0 0 0a²x x ' acx y' abx ' y bcy y' 1

0 0 0 0 0 0 0 0a(x x ' cx y' bx ' y ) bc(y y' ) 1

0 0 0 0 0 0u x x ' cx y' bx ' y

0 0v y y'

*nnPGCD(a;b ) 1

n 1PGCD(a;b) 1nPGCD(a;b ) 1n 1PGCD(a;b ) 1

nPGCD(a;b ) 1PGCD(a;b) 1n 1PGCD(a;b ) 1 4

19541962196219541954PGCD(981 ;977) 11954 1962PGCD(981 ;977 ) 11954 8PGCD(981 ;2 ) 1

41954 1962 1954 1954 1962 8 1954PGCD(1962 ;1954 .) 2 .PGCD(981 ;977 .2 ) 2

3- PGCD(2013;1962)

2013 3 671 1962 3 654 PGCD(2013;1962) 3PGCD(671;654)

PGCD(671;654) 1

PGCD(2013;1962) 3

(E)

2013x 1962y 54: (E)

3211332013x3196231962y

32013x 1962y354(E)

(x;y)(E) x 0 6

2013x 1962y 54: (E) 671x 654y 18 (E)3

671x 654y 18 671x 654y 18 671x 6(109y 3)

671x 0 6 x 0 66716

0 0(x ;y )

074 x 80(E)

0x 0 6

074 x 80

0x 78

0x 78

0y 80

0x 78671x 654y 18

(78;80)671x 654y 18

6:42363

4242 78

671x 654y 18...(1)

671(78) 654(80) 18.....(2)

21

671(x 78) 654(y 80) 0 671(x 78) 654(y 80).....(*)

(*)671654(y 80)671(y 80)671654

(y 80) 671k y 671k 80 y(*)x 654k 78

(x;y)(x;y) (654k 78;654k 80) k

2d

dxy(x;y)(E)

dxd671xdd654y

d671x 654yd1818d D 1;2;3;4;6;9;18

ab

(a;b)671a 654b 18 PGCD(a;b) 18

a 0 18

b 0 18

654k 78 0 18

671k 80 0 18

17k 2 0 18

k 2 18 k 2 18k 18p 2 p

k 18p 2 (a;b) (11772p 1386;12078p 1422)

1PGCD( ; ) PGCD( ;10)

PGCD(a;b) PGCD(b;r)

a b.q r qra b

32n 14n 2 n 3

(2n² 6n 4) ( 10)

PGCD( ; ) PGCD( ; 10 ) PGCD( ; ) PGCD( ;10)

PGCD( ; )

PGCD( ; ) d PGCD( ;10) d d11

10d D 1;2;5;10

nPGCD( ; ) 5

PGCD( ; ) 5 PGCD( ;10) 5 0 5 n 3 5k

PGCD(5k;5.2) 55PGCD(k;2) 5PGCD(k;2) 1

6;42353

4242 79

kk 2p 1 n 5(2p 1) 3 n 10p 2 p

0 5 n 3 0 5 n 5k 3 k

4n411

04 1 11 14 4 11 24 5 11 34 9 11 44 3 11 54 1 11n4115

5k 45k 35k 25k 15kn

1139541n4

n

5n n4 4 n 0 11

n 2 10

5n n4 4 n 0 11

n 2 10

5p 1 5 5p 1(4 ) (4 )² n 0 11

n 10p 2

5p 14 4 11 5p 1(4 )² 5 11 5p 1 5(4 ) 1 11

5p 1 5 5p 1(4 ) (4 )² n 0 11

n 10p 2

n 5 11

n 2 10

n 6 11

n 2 10

10n 60 110

11n 22 110

n 82 110

n 110 82

3n 2n 27 0 n 1

2n 27 0 n 1 2n 2 25 0 n 1 25 0 n 1

25 0 n 1 n 12525n 1 D 1;5;25

n 1 1;5;25 n 0;4;24

(a;b)(b a)(a b) 24

(a;b)24324 2 3 (3 1)(1 1) 8

24D 1;2;3;4;6;8;12;24

(b a)(a b) 24 (a;b)

1(b a)(a b) 2 12 (a b) 12

(b a) 2

(a;b) (5;7)

7242355

4242 7:

2(b a)(a b) 4 6 (a b) 6

(b a) 4

(a;b) (1;5)

24

(b a)(a b) 24 b² a² 24 b² a² ( 24)²

1(a;b) (5;7)( 24)²7² 5²

24ABABCAAC 5BC 7

2(a;b) (1;5)( 24)²5² 1²

24ABABCAAC 1BC 5

2

10141 4 3 2 1 01.5 0.5 1.5 4.5 1.5 671

3403 3 2 1 03.5 4.5 0.5 3.5 478

(a;b)²b² a² 24

a b 9

b² a² 24

a b 9

b² a² 24

671a 478b 9

1(a;b) (5;7)b² a² 24

671a 478b 9

2(a;b) (1;5)b² a² 24

671a 478b 9

b² a² 24

a b 9

(a;b) (5;7)

321131434

2013 3.6711434 3.478

211314343

671478

PGCD(2013;1434) 3.PGCD(671;478)

671671 25,967123

4242 7;

PGCD(671;478) 1

²(x;y)2013x 1434y 27

2013x 1434y 27 671x 478y 9

2(5;7)671x 478y 9

671x 478y 9...(1)

671(5) 478(7) 9.....(2)

671x 478y 9...(1)

671(5) 478(7) 9.....(2)

21671(x 5) 478(y 7) 0

671(x 5) 478(y 7).....(*)

(*)478671(x 5)478(x 5)671478

(x 5) 478k x 478k 5 x(*)y 671k 7

(x;y)(x;y) (478k 5;671k 7) k

34233

211162011 44,8444

0 0(x ;y )11

2011 1432 1 579

1432 579 2 274

579 274 2 31

31 579 274 2

31 579 274 (1432 2 579)

31 579 2 (1432 2 579)

31 1432( 2) 5 579

31 1432( 2) 5 (2011 1432)

31 2011(5) 1432(7)

(5;7)1

2011x 1432y 31...(1)

2011(5) 1432(7) 31.....(2)

2011x 1432y 31...(1)

2011(5) 1432(7) 31.....(2)

212011(x 5) 1432(y 7) 0

2011(x 5) 1432(y 7).....(*)

(*)14322011(x 5)1432(x 5)14322111

(x 5) 1432k x 1432k 5 x(*)y 2011k 7

(x;y)(x;y) (1432k 5;2011k 7) k

7342343

4242 82

2nn27

02 1 7 12 2 7 22 4 7 32 1 7n273

3k 23k 13kn 421n2

2012143220117

2011 2 7 2012 20121432 14322011 2 7

1432 1 3 20121432 1 3201214323k 1

20121432 3k 12011 2 7

201214322011 2 720121432201172

n n n n2010 2011 1432 0 7

2011 2 7 n n2011 2 7 2010 1 7 n2010 1 7

1432 4 7 n n1432 (2 )² 7

n n n2010 2011 1432 0 7 n 2n1 2 2 0 7

n

nn 3k 1 n 3k 2

N

N29

( ; ) 12011 1432 31

x 1432k 5 y 2011k 7 0 8 0 8

0 1432k 5 8 0 2011k 7 8 k 0

5 7

3k 23k 13kn 421n2 2412n2 1132n n2 2 1

4242 83

32 3

3 0N 2 2735 2.9 7.9² 3.9 5.9 2057

1(E)

13x 7y 1...(E) x y

(E)

(E)13x 7y 1 13x 1 7 x 1 7 13 1 7

x 1 7x 7k 1 k

x13x 7y 1 y 13k 2

(x;y) (7k 1;13k 2) k

2a

a 1 7

a 0 13

a 1 7

a 0 13

13a 13 91

7a 0 91

6a 13 91 a 13 91

a 91p 13 p

(E)

3nn9713

nn97

09 1 7 19 2 7 29 4 7 39 1 7n973

3k 23k 13kn

7421n9

nn913

09 1 13 19 9 13 29 3 13 39 1 13n9133

3k ' 23k ' 13k 'n

13391n9

4 b91

6 3b 00 086 .9 .9 8.9 6

7442333

4242 84

b91

b 0 7

b 0 13

91 7.13713

1 0 7

0 13

69 0 7 39 0 7 69 0 13 39 0 13 78 1 7

291p 13

0 8 0 8 0 91p 13 16 p 0

b91

( ; ) (5;8) ( ; ) (8;5) ( ; ) (6;7) ( ; ) (7;6)

1n3n3 113

33 1 13n 3n3 1 13

n3n3 113

2 n3n 13 3 3n 23 9 13

n 3n3 1 13 3n 13 3 13 3n 13 3 0 13

n 3n3 1 13 3n 23 9 13 3n 23 9 0 13

3 n n313

03 1 13 13 3 13 23 9 13 33 1 13n3133

3k ' 23k ' 13k 'n

13931n3 2010200513

2005 3 13 2010 0 3 2010 3.6702005 3 13 20102005 1 1320102005131

4p

A13p 3n

p 2 p 3p

pA 3 3 3

3n 2(3n ) 3(3n )

3nA 3 3 3

3n3 1 13 2(3n )3 1 13 3(3n )3 1 13 3nA 3 13

p 3n 1 p

A133n 1 2(3n 1) 3(3n 1)

3n 1A 3 3 3

3n 13 3 13 2(3n 1)3 9 13 3(3n 1)3 1 13 3n 1A 0 13

7542323

4242 85

pA13p 3n 2

3n 2 2(3n 2) 3(3n 2)

3n 2A 3 3 3

3n 23 9 13 2(3n 2)3 3 13 3(3n 2)3 1 13 3n 2A 0 13

pA13p 3n 2 1

abp

A

9 6 3a 1001001000 1.3 1.3 1.3 3

a A

12 8 4b 1000100010000 1.3 1.3 1.3 4

b A

ab13

3a A 3

A 3 13 a 3 13a133

4b A 4

A 0 13 b 0 13b130

1(x;y)1y7

7x 65y 2009....(1) 65y 2009 7x 65y 7(287 x)

65y 0 7 y 0 7657

y 0 7y7

1

y7y 7kk

y65y 7(287 x) x 65k 287

(x;y)(x;y) ( 65k 287;7k) k

2nn29

02 1 9 12 2 9 22 4 9 32 8 9 42 7 9 52 5 9 62 1 9n296

6k 56k 46k 36k 26k 16kn 578421n2

3n6n2 3n 2 9

6n2 3n 2 9 6n2 3n 2 0 9

6n2 3n 2 0 9 1 3n 2 0 9 3n 3 9

n 2 3n 3p 2 p

7642324

4242 86

4n

u9

6n

nu 2 1

2 6n2 1 9 6n2 1 0 9 nu 0 9

nu9

1 2(7u )x (u )y 126567....(2)

1 2(7u )x (u )y 126567....(2) 7x 65y 2009....(2)

1u 1 9 2

u 65 9 126567 2009 9

21

0 0(x ;y )2

0x

0y

0y 25

0y 25

0x 07k 2565k 287 0

25 287k

7 65

k 40 0

(x ;y ) (27;28)

1(5x² 6)(x 1) xy3(5x² 6)(x 1) 5x 5x² 6x 6 A....(1)

A (5x² 6)(2 2y).....(2)

122 2y x 1 1 2y x

xyx12

x712x711

1 2y x y 3x 7

y 5x 11

A

1x 7y 3A (5.7² 6)(2 2.3) 2008

1x 11y 5A (5.11² 6)(2 2.5) 7332

25842.584 1².2 .2.73

58412

77422;3

4242 87

aba ba b 32

a² b² 584

PGCD(a;b) da d.a 'b d.b'PGCD(a ';b') 1

a b 32

a² b² 584

32a ' b '

d

584a '² b '²

d²

d 1d 2

1d 1a ' b ' 32

a '² b '² 584

b' 32 a '

a '² 32a ' 220 0

(10;22)(22;10)1122

1d 2a ' b ' 16

a '² b '² 146

b' 16 a '

a '² 16a ' 55 0

(11;5)(5;11)511

ab(a;b) (22;10)

1(E)²

3x 21y 78....(E) PGCD(3;21) 3

3x 21y 78....(E) 378

(x;y)²(E) x 5 7

(E)x 7y 26....(E') x 26 7y

x 26 7y x 26 7 x 5 7 26 5 7

(E)

x 5 7x 7k 5 k

x 7k 5 x 7y 26....(E') y k 3

(x;y)(x;y) (7k 5;k 3) k

2nn57

05 1 7 15 5 7 25 4 7 35 6 7 45 2 7 55 3 7 65 1 7n576

6p 56p 46p 36p 26p 16pn 3326451n5

78422:4

4242 88

(x;y)²(E) x y5 5 3 7

(E)(x;y)²(x;y) (7k 5;k 3) k3

k ' k 3 k'

(E)(x;y)²(x;y) (7k ' 21;k ')

x y5 5 3 7 7k ' 26 k '5 5 3 7

7k ' 26 k '5 5 3 7 6k ' k ' 6.4 2 k '5 .5 .5 5 3 7

k ' 15 3 7 6k '5 1 7 6.4 25 4 7

k ' 1 6p 5 k ' 6p 4

k'(x;y) (42p 54;6p 4) p