I
1 1.0 ( ) ( ) 1 2 2
3 4 1.1 ( ) 1, 2, 3, 0 1 ( ) (1) 0 (2), 3, 2, 1, 0, 1, 2, 3, 0, 1, 2, 3, 2 (3) 5 N 3 ( ) ( ) 1 ( ) 2 3, 3, 2, 1, 0, 1, 2, 3, 3 Z 3
? 374 14 26 10 ( ) ( ) + ( ) ( ) = 14, ( ) = 26, ( ) = 10 ( ) ( ) + ( ) = 14 26 + 10 = 374 ( ) = ( ) ( ) + ( ) ( ) 1 ( ) = ( ) ( ) + ( ) (1) 34 3 (2) 124 11 (3) 225 25 ( ) a b q r 4 a = bq + r ( 0 < = r < b) 1 4 q r quotient remainder 4
1.2 0 ( ) a b 0 a = bq a b b a b a a b ( ) 24 4 (!) 6 24 = 4 6 4 24 24 4 ( ) a a = 1 a 1 0 = 0 a 0 1.3 6 6 = 2 3 5
2 6 6 = 3 2 3 6 2 6 3 6 2 3 = 3 2 6 = 2 3 2 3 6 ( 2 ) ( 3) 6 6 2 3?! 6 = 1 6 1 6 6 6 1, 2, 3, 6? 3 6 4 5 6 6 6 1, 2, 3, 6 ( ) 16 16 = 1 16 1 16 16 = 2 8 2 8 2 3 16 4 16 = 4 4 6
a a > = 4 b 16 = a b b 4 5 4 a 4 16 1 2 4 8 16 ( ) 2 (1) 12 (2) 28 (3) 60 (4) 29 a a = a 1 1 a a 1 1574? 1.4 1.4.1 7 2 (4) 1 29 29 1 7
( ) 1 6 1 ( ) 1 1 ( 1) 29 10 2 1 2 2 3 1 3 3 4? 4 1 2 4 2 5 7 1 10 2 3 5 7 10? 2 10 20 ( ) 1 20 1 5 b > 4 a b > 16! a b = 16!! 6 8
20 7? 2( (629 )) 20 1.4.2 30 30 30 = 2 3 5 30 2 3 5 1 ( ) 1 ( ) 8 ( ) 2 3 5 = 30 30 30 = 2 3 5 7 8 a = p q p a q a 9
( ) 30 30 2 9 30 = 2 15 15 3 15 = 3 5 5 30 30 = 2 3 5 ( ) 2 ) 504 2 ) 252 2 ) 126 3 ) 63 3 ) 21 7 504 504 = 2 2 2 3 3 7 9 20 10
2 504 = 2 3 3 2 7 ( ) 3 (1) 45 (2) 60 (3) 180 (4) 1024 (5) 29 30 5 ( ) 10 1.4.3 6 12 10 6 = 2 3 = 1 2 3 = 1 1 2 3 = ( ) 11
12 12 = 2 2 3 6 6 = 2 3 12 2 2 3 6 2 3 11 6 12 8 12 8 8 = 2 3 2 8 12 3 2 12 ( ) ( ) a b a b a b b = aq ( ) a q b = a q b a b ( ) 11 12 2 2 3 12
( ) a b a b 12 a b b a b = a a = a = a = q b = aq a b ( ) 504 = 2 3 3 2 7, 36 = 2 2 3 2 36 504 2 7 504 = (2 2 3 2 ) (2 7) 504 = 36 14 36 504 ( ) 4 792 (1) 12 (2) 34 (3) 63 (4) 72 (5) 132 12 A B B A A B A B B A A B B A 13
(0 ) a( 0) a 0 = 1 ( ) 72 72 72 = 2 3 3 2 2 3 2 3 2 3 2 p 3 q (p, q) (0, 0), (1, 0), (2, 0), (3, 0) (0, 1), (1, 1), (2, 1), (3, 1) (0, 2), (1, 2), (2, 2), (3, 2) 12 13 2 0 3 0, 2 1 3 0, 2 2 3 0, 2 3 3 0 2 0 3 1, 2 1 2 1, 2 2 3 1, 2 3 3 1 2 0 3 2, 2 1 3 2, 2 2 3 2, 2 3 3 2 13 14
1, 2, 4, 8 3, 6, 12, 24 9, 18, 36, 72 12 72 72 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 ( ) ( ) 5 (1) 20 (2) 63 (3) 256 (4) 648 1 300 300 300 = 2 2 3 5 2 2 3 2 3 5 2 0 2 a 3 b 5 c (a, b, c) (0, 0, 0), (0, 1, 0) (0, 0, 1), (0, 1, 1) (0, 0, 2), (0, 1, 2) 2 1 (1, 0, 0), (1, 1, 0) (1, 0, 1), (1, 1, 1) (1, 0, 2), (1, 1, 2) (a, b, c) a 3 2 2 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300 ( ) 6 84 15
1.5 2 3 3 1.5.1 2 ( ) 5 2 5 1 0 2 4 6 8 2 0 5 5 (2 ) 1 0 2 4 6 8 (5 ) 1 0 5 5 2 10a + 2b b 0 1 2 3 4 14 10a + 2b = 2(5a + b) 2 ( ) 2 ( ) 4 2 5 5 14 a 2b 346 a = 34, b = 3 a b 16
1.5.2 4 8 4 8 4 8 (4 ) 4 00 4 100a + b ( b ) 4 b = 4k (k ) 100a + b = 4(25a + k) 4 ( ) 1024 24 4 1024 4 123456789 89 4 123456789 4 ( ) 7 4 (1) 172 (2) 122 (3) 2348 (4) 2345362 (5) 342233432 (8 ) 8 000 8 5 123456 456 8 8 123454 454 8 8 ( ) 8 8 (1) 172 (2) 122 (3) 2348 (4) 2345362 (5) 342233432 17
1.5.3 3 9 3 9 (3 9 ) 3 3 9 9 a, b, c, d 1000a + 100b + 10c + d 15 9 a + b + c + d 9 1000a + 100b + 10c + d = 999a + 99b + 9c + (a + b + c + d) = 9(111a + 11b + c) + (a + b + c + d) 9(111a+11b+c) 9 ( ) 9 a+b+c+d 9 9(111a+11b+c)+(a+b+c+d) 9 9 16 1000a + 100b + 10c + d 9 ( ) 6 3 2346 2 + 3 + 4 + 6 = 15 3 2346 3 1357 1 + 3 + 5 + 7 = 16 3 1357 3 ( ) 15 2346 1000 100 10 ( a = 2, b = 3, c = 4, d = 6 1000 = 10 3, 100 = 10 2, 10 = 10 1, 1 = 10 0 10 3 a + 10 2 b + 10 1 c + 10 0 d p 16 9 9 9 9n, 9m 9n + 9m 18
1.5.4 6 6 6 2 3 6 ( ) a 2 3 2 a = 2k 2 3 k 3 k = 3l a = 2 (3l) = 6l a 6 ( ) 1.5.5 9 87, 136, 275, 635, 1080, 71928 (1) 3 (2) 6 (3) 9 (4) 4 1.5.6 1 7 11 11 7 1.6 19
1.6.1 ( ) 2 ( ) 12 1, 2, 3, 4, 6, 12 18 1, 2, 3, 6, 9, 18 12 18 1, 2, 3, 6 ( ) 10 (1) 24, 40 (2) 12, 36 (3) 40, 72, 96 (4) 14, 33 1, 2, 3, 6 6 ( ) ( ) 8 ( ) 20
a, b a b d d a b a = pd, b = qd 17 d a b d a b d a b a = p d, b = q d d a b d > = d d = rd d d ( ) a b d a b d a b d a b a b 72 84 72 = 2 2 2 3 3 84 = 2 2 3 7 17 21
2, 2, 3 2 2 3 = 12 ( ) 2 ) 72, 84 2 ) 36, 42 3 ) 18, 21 6, 7 2, 2, 3 48, 36, 90 2 3 = 6 2 ) 36, 48, 90 3 ) 18, 24, 45 6, 8, 15 ( ) 11 (1) 24, 40 (2) 12, 36 (3) 40, 72, 96 1.6.2 22
( ) 1 ( ) 12 35 1 12 35 ( ) 12 (1) 6, 8 (2) 15, 49 (3) 48 60 ( ) b a a b ( ) ( ) b a b a a b d 1 a = a d, b = b d a b b a b a = b d a d = b a ( ) 12 18 12 18 6 (!) 12 18 = 2 6 3 6 = 2 3 ( ) 13 (1) 24 (2) 12 (3) 56 40 36 432 23
1.6.3 ( ) a, b a b q r a b b r a b d a = a d, b = b d 18 a b 1 9 a b ( ) a b q r a = bq + r r r = a d b qd = (a b q)d b = b d d d b r d d b r 19 b b r a b q 1 d b = b d, a b d = cd d > 1 a b q = cd a = b q + cd = b d q + cd = (b q + c)d 18 19 24
a = a d = (b q + c)(dd ) b = b d = b (dd ) dd a b d > 1 dd > d d a b d b r ( ) 2 12936 9828 12936 9828 3108 12936 9828 9828 3108 9828 3108 3 504 9828 3108 3108 504 12936 = 9828 1 + 3108 9828 = 3108 3 + 504 3108 = 504 6 + 84 504 = 84 6 84 ( ) ( ) 14 (1) 56, 432 (2) 16632, 10010 1.6.4 ( ) 2 ( ) 25
4 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 6 0, 6, 12, 18, 24, 30, 36, 42, 4 6 0, 12, 24, 36, ( ) (1) (2) 0 = a 0 0 ( 1 ( ) 15 0 (1) 2, 3 (2) 6, 9 (3) 7, 14 (4) 12, 18 (5) 48, 72 ( ) ( ) 10 ( ) a, b a b d a = pd, b = qd p q l = pqd l = (pd)q = aq, l = (qd)p = bp l a b l a b l = l a = l pd 26
l b l q p q l a b l = qr l = l pd = r(qpd) = rl r > = 1 l > = l l l l ( ) ( ) a, b d a = a d, b = b d a b a b d 72 84 72 = 2 2 2 3 3 84 = 2 2 3 7 12 p, q 6, 7 12 6 7 = 504 72 72 = 2 2 2 3 3 84 84 = 2 2 3 7 2 2 2 3 3 7 ( ) 2, 2, 3 6, 7 27
2 ) 72, 84 2 ) 36, 42 3 ) 18, 21 6, 7 16 (1) 9, 12 (2) 24, 36 (3) 32, 80 36, 48, 90 36 = 2 2 3 3 48 = 2 2 2 2 3 90 = 2 3 3 5 2 2 2 2 3 3 5 = 720 ( ) 11 2 ) 48, 36, 90 3 ) 24, 18, 45 2 ) 8, 6, 15 3 ) 4, 3, 15 4, 1, 5 2 3 2 3 4 1 5 28
17 (1) 12, 18, 24 (2) 7, 13, 15 (3) 28, 42, 70 (4) 126, 196, 378 1.6.5 (!) 5 12 + 7 18 ( 12 18) OK 1.6.6 ( 20 ) a, b d l ab = dl 12 3 a, b 21 546 a b 20 29
21 a = 21p, b = 21q p q ab = 21 546 21p 21q = 21 546 21 21 pq = 26 26 = 2 13 p q p = 2, q = 13 p = 13, q = 2 p = 1, q = 26 p = 26, q = 1 a 21 a = 21p, b = 21q p q ab = 21 546 21p 21q = 21 546 pq = 2 13 p q p = 2, q = 13 p = 13, q = 2 p = 1, q = 26 p = 26, q = 1 a p = 2, q = 13 p = 26, q = 1 a = 21 2, b = 21 13 a = 21 26, b = 21 a = 42 b = 273 a = 546 b = 21 ( ) ( ) 30
1.7 ( 18 19 300 ( ) n > = 3 x n + y n = z n 31
(x, y, z) 3 1989 89 1989 1 6 1976 1997 18 32
1 2 0 33