bac2020...n 12 41{> @ 3 b 6n² 19n 15 a 20n² 36n 9 2n 3 b a n n k 1 4 1 115k {> @ 2 n 11 4n 3...
TRANSCRIPT
**************************************
...202 00 144
**************************************
( ;14PGCD 40)2020 ...
BAC2020
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
,a b
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
13n7
123456n
23n7n
37 2020 1440 3 23 10 9 n
1n1101101
n214
251 13 35
) ,x y45 28 130x y
xy5
n2 395 6
7
9n1271n x
1xn8
2xn11
x (1 y
yxxy7
1(5)
413a (3)
102b
ab
7a ba b
2-x( ) ( ) ( )
12 34 452x x x
(9) (11)
52xxx
25
26
29
27
28
10
10291010 4
11,x y241 27 1x y
1 ( أ
1 ( ب
241 27 5....... 2x y22
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
, 12 gcd ,
105
ppcm x y p x y
x y
x y
3
2 2
3 ,
405
ppcm x y xy
x y
4
( , ) 100
gcd( , ) 5
ppcm x y
p x y
1n1n2 3n
1n3 4n
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
nn n n n n
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
1nn1013
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
1104x 20y 272....(E) (x;y)xy
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
nnun 1u
2n(v )nn n
1v u
3
n(v )0v
nnSn 0 1 2 3nS v v v ..... v
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
n5n 4 10n 42 2016 1437 11
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
, ,O i j D
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
» Fermat«pa
pp1 1pa
nu0 1u n1 10 21n nu u
1 1u2u3u
2 n13 10 7n
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
( )nu 0 1u n1 10 81n nu u
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
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