0 http://homepage3.nifty.com/yakuikei (18) 1 99 3 2014/12/13 (19) 1 100 3 n Z (n Z ) 5 30 (5 30 ) 37 22 (mod 5) (20) 201 300 3 (37 22 5 ) (12, 8) = 4 (21) 16! 2 (12 8 4) (22) (3 n )! 3 (23) 100! 0 1 (1) 2 < n 7 n 3 (2) n 2 20 n (3) 1 30 (4) 0 30 (5) 10 30 2 (6) 17 5 (7) 17 5 (8) 2 5 2 (9) 12 (10) 3 n n 2 50 (11) 75228 2 (12) 75228 4 (13) 75228 8 (14) 75228 5 (15) 75228 3 (16) 75228 9 (17) 75228 6 3 (24) p (25) 50 (26) 2011 (27) 56 (28) 361 (29) 3600 (30) 363 (31) 364 (32) 365 (33) 1001 (34) ab = 17 a, b (a, b) (35) (n + 1)(n + 3) (n ) 5 n 5 2 4 (36) m 2 = 5n + 1 m 5 1 or 4 4 (37) 200 (38) 200 (39) 1800 N (40) 1800 S
1 (41) p m q n (p, q ) (57) 6 n(n + 1)(n + 2) N (58) 6 n(n + 5)(n + 10) (42) p m q n S 5 GCD LCM (59) n 3 n (60) n 5 n (mod 3) (mod 30) (43) 24 36 GCD LCM (44) 2700 525 GCD LCM (45) 2 a, b GCD LCM G, L ab = GL (61) a 1, b 4 (mod 6) a + b, 3a 2b, ab 6 (62) 6 n 3 + 5n 8 (63) 4 n 3 6 (46) a = bq + r (a, b) = (b, r) ( ) (47) 7337 3451 (48) 5423 1911 (49) n + 1 n 1 (50) n 2 + 1 n + 1 7 ( ) (64) 2 n 3 (65) n 4 2 n 1 (mod 15) (66) 8 n 2 n 1 (mod 17) (67) 3 5 n 2 n (n N) (68) 30 5 n 3 n 2 n (n : odd) (69) 7 3 3n 2 + 5 3n 1 (n N) 9 (70) n 2 3 (71) n 2 4 (72) n 2 3 (mod 6) n 3 (mod 6) (51) 9876 9855 7 (52) n 3 1 2n, n 2, n 3, n 9 3 (53) n 3 2 2n, n 2, n 3, n 9 3 (54) 2 n(n + 1) (55) 2 n(n + 11) (56) 3 n(n + 1)(n + 2) (73) a 2 + b 2 3 a, b 3 (74) 7 a 2 + b 2 = 7 a, b 10 (75) 2k + 1 2k 1 (76) 8x = 9y 9 x (77) 7 7 C k (k = 1, 2, 3,..., 6) (78) 8a 8b (mod 9) a b (mod 9)
2 (79) n(n + 1)(n + 2)(n + 3) 12 (98) n 2 + 3 N n( N) 12 p (80) xy = n 2 (n N) x, y x, y (99) 2 N = 1101 (2) 10 (81) a 2 = (b + c)(b c) (b, c : odd, b c (100) 10 N = 100 2 ) b = m 2 + n 2, c = m 2 n 2 (101) 6 N = 454 (6) 3 11 (102) 7 n M 1 7 (82) 5x = 4y (103) 7 2 (83) 20x = 16y (84) 4x 3y = 5 (85) 5 2 7 1 (86) 37x + 13y = 1 (x, y) 1 (87) x 2 4y 2 = 5 (88) xy 3x 2y = 0 (x, y N) A =a n a n 1 a 1 a 0(7) =a n 7 n + a n 1 7 n 1 + + a 1 7 1 + a 0 1, B =b n b n 1 b 1 b 0(7) =b n 7 n + b n 1 7 n 1 + + b 1 7 1 + b 0 1 (a 0, a 1, ; b 0, b 1 0 6 ) ( ) (a 0, a 1,, a n 1, a n ) = (b 0, b 1,, b n 1, b n ) A = B 13 (89) x 2 y 2 + 2y 13 = 0 (90) x 2 + 4y 2 = 5 (104) (a n ) a 1 = 1 a 2 = 1 a n+2 = a n+1 +a n 3 n = 2 a n (91) x 2 + y 2 = 85 (92) 3x 2 + 2xy + y 2 = 11 a 3 =a 2 + a 1 = 1 + 1 = 2. 1 (93) abc = bc + ca + ab (1 a b c) (94) abc = a + b + c (1 a b c) (95) 2 x (2y + 1) = 2000 x, y (96) m 2 = 2 n + 1 (n > 0) (97) 2n + 6 n 2 + n N n( N) a n+3 =a n+2 + a n+1 =(a n+1 + a n ) + a n+1 =2a n+1 + a n. i.e. a n+3 a n = 2a n+1. 2 a n+3 a n. i.e. a n+3 a n (mod 2). 2 1 2 a 3, a 6, a 9, a 12,... : even.
3 (105) (a n ) a 1 = 2 a 2 = 3 a n+2 = 5a n+1 + a n (n = 12, 3, ) 1 2 n a n a n+1 15 (106) 2 (107) a + b 2 = 0 (a, b Q) a = 0, b = 0 2 / Q (108) a + b 3 = a + b 3 (a, b, a, b Q) a = a, b = b 3 / Q (109) a, b, c, d Z x 3 ax 3 + bx 2 + cx + d = 0 ( ) p q (p, q q > 0) p d, q a
4 1 (1) n = 1, 0, 1, 2. (2) n = 0, ± 1, ± 2, ± 3, ± 4. (10) (11) n = 6, 3, 0, 3, 6. 3 0 ( ) 7 ITEM1 8 2 75228 2 (3) 30. 1 (4) 1 30 30 0 31 (5) 30 {}}{ 1, 2,..., 9, 10, 11,..., 30 }{{}}{{} 9 1 30 30 1 9 9 10 30 30 9 = 21 1 () ( 1). (12) 28 4 75228 4 (13) 228 8 75228 8 (14) 8 5 75228 5 (15) 7 + 5 + 2 + 2 + 8 = 24 3 75228 3 (16) 24 9 75228 9 (17) 75228 2 3 6 (= 2 3) 2 3 () (6) 17 = 5 3 + 2. 3, 2. (7) 17 = 5 ( 4) + 3. 4, 3. (8) 2 = 5 0 + 2. 0 (18) (1, 2, 3), (4, 5, 6),..., (97, 98, 99) 3 3 1 99 = 3 33 33 3 33 0, 2. 2 (9) 1 2 3 1 2 3 12 6 4 12 6 4 12 (19) (1, 2, 3), (4, 5, 6),..., (97, 98, 99), 100 3 3 100 3 100 = 3 33 + 1 33 3 33
5 (20) 1 300 3 300 = 3 100 100 1 200 3 200 = 3 66 + 2 66 (23) 10 = 2 5 0 100! 2 m 5 n 2 < 5 m n n = 100 5 + 100 5 2 = 20 + 4 = 24. 201 300 3 100 66 = 34 201 300 300 200 (25) = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. 100 3 (21) 16! = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 N 1 16 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 16 2, 4 = 2 2, 8 = 2 3, 16 = 2 4 2 16 2 4 16 4 8 16 8 16 16 16 = 8, = 4, = 2, = 1. 16! 2 1 N N = 16 2 + 16 2 2 + 16 2 3 + 16 2 4 =8 + 4 + 2 + 1 = 15. (24) p 1, p 2 1 1 1 1 2 3, 5, 7, 9,... 1 (26) 2011 = ab (a, b 2 a b) a 2 ab = 2011 a 2011. 2011 2011 44 2 }{{} 1936 < 2011 < 45 2 }{{} 2025 44 44 < 2011 < 45. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 2011 2011 (27) 56 = 8 7 = 2 3 7. () (28) 361 = 19 2. ( 19 2 = 361 ) (22) 3 n 3 + 3n 3 2 + 3n 3 3 + + 3n 3 n =3 n 1 + 3 n 2 + + 3 2 + 3 + 1 =1 3n 1 3 1 = 1 2 (3n 1). (29) 3600 = 60 2 = (2 2 3 5) 2 = 2 4 3 2 5 2. ( ) (30) 363 = 3 121 = 3 11 2. ( 3 11 2 = 121 )
6 (31) 364 = 4 91 = 4 7 13. (38) ( 2 64 4 ) 1 1 + 1 5 + 1 5 2 (32) 365 = 5 73. 1 +2 1 + 2 5 + 2 5 2 ( 5 ) +2 2 1 + 2 2 5 + 2 2 5 2 +2 3 1 + 2 3 5 + 2 3 5 2 (33) 1001 = 7 143 = 7 11 13. =1 (1 + 5 + 5 2 ) ( 2, 3, 5 7 ) 1000 = 7 143 1 +2 (1 + 5 + 5 2 ) +2 2 (1 + 5 + 5 2 ) +2 3 (1 + 5 + 5 2 ) 10 3 1 (mod 7). = ( 1 + 2 + 2 2 + 2 3) ( 1 + 5 + 5 2) 10 6 ( 1) 2 = 1 (mod 7). =15 31 = 465. (34) 17 (a, b) (39) 1800 = 18 100 = 2 3 2 (2 5) 2 = 2 3 3 2 5 2. (1, 17), (17, 1), ( 1, 17), ( 17, 1). N = (3 + 1)(2 + 1)(2 + 1) = 4 3 3 = 36. (35) 5 5 (n + 1)(n + 3) 5 n + 1 5 n + 3. 5 n + 1 k n + 1 = 5k, i.e. n = 5k 1 = 5(k 1) + 4 n 5 4 5 n + 3 n + 3 = 5k, i.e. n = 5k 3 = 5(k 1) + 2 n 5 2 (36) 5n =m 2 1 =(m 1)(m + 1). 5 5 m 1 or 5 m + 1. m 1 or 1 (mod 5). i.e. m 1 or 4 (37) 200 (mod 5). 200 = 2 10 2 = 2 (2 5) 2 = 2 3 5 2. 200 ( ) i = 0, 1, 2, 3 2 i 5 j. j = 0, 1, 2 (3 + 1)(2 + 1) = 4 3 = 12 (40) S =(1 + 2 + 2 2 + 2 3 )(1 + 3 + 3 2 )(1 + 5 + 5 2 ) =15 13 31 = 6045 (41) N = (m + 1)(n + 1). (42) S = ( 1 + p + p 2 + + p m) ( 1 + q + q 2 + + q n) }{{}}{{} =1 pm+1 1 p 1 = pm+1 1 p 1 (43) GCD 1 qn+1 1 q 1 qn+1 1 q 1 24 24, 12,... 36 12,.... GCD 12 LCM 36 36, 72,... 24 72,.... LCM 72 (44) 2700 = 27 100 =3 25 36 525 = 25 21 =3 25 7 36 7.
7 (47) () (45) GCD =3 25 = 75, LCM =3 25 36 7 = 18900. 2700 =2 2 3 3 5 2, 525 = 3 5 2 7. GCD 3 5 2 = 75. LCM 2 2 3 3 5 2 7 = 18900. { a = Ga, (a, b ) b = Gb 7337 = 3451 2 + 435, 3451 = 435 7 + 406, 1 435 = 406 1 + 29. 406 = 29 14 + 0. (7337, 3451) =(3451, 435) =(435, 406) =(406, 29) =(29, 0) = 29. 1 0 = k 0 0 k k 0 L =Ga b. ab =Ga Gb =G Ga b. ab =GL. GCD (46) a b S,b r T S = T 2 1 (48) () d b, d r ( ) d a T S. 3451 = 435 8 29, 435 = 29 15 + 0. (7337, 3451) =(3451, 435) =(435, 29) =(29, 0) = 29. d a, d b ( ) r = a bq d r S T S = T. 12 2 b r ( ) q, r a b 5423 = 1911 2 + 1601, 1911 = 1601 1 + 310, 1601 = 310 5 + 51. 310 = 51 6 + 4. 1 51 = 4 12 + 3. 4 = 3 1 + 1.
8 (50) (5423, 1911) =(1911, 1601) =(1601, 310) =(310, 51) =(51, 4) 1 =(4, 3) =(3, 1) = 1. 1 1 2 r 1 1 2 1 1 1 n 2 + 1 = (n + 1)(n 1) + 2. 1 (n 2 + 1, n + 1) = (n + 1, 2) = (51) 9876 9855 = 21 = 7 3 7 9876 9855. (52) n = 3k + 1 (k ) 2n =2(3k + 1) = 3 2k + 2. 2n 3 2. { 2 (n : odd), 1 (n : even). 2 5423 = 1911 3 310, 1911 = 310 6 + 51, 310 = 51 6 + 4. (5423, 1911) =(1911, 310) =(310, 51) =(51, 4) = 1. (49) n + 1 = (n 1) 1 + 2. 1 1 { 2 (n : odd), (n + 1, n 1) = (n 1, 2) = 1 (n : even). 1 ( ) n 1 2 () > ( ) n 2 =(3k + 1) 2 =(3k) 2 + 2 3k + 1 = 3(3k 2 + 2k) + 1. n 2 3 1. n 3 =(3k + 1) 3 =(3k) 3 + 3(3k) 2 + 3 3k + 1 =3(9k 3 + 9k 2 + 3k) + 1. n 3 3 1. n 9 =(3k + 1) 9 =(3k) 9 + 9 C 1 (3k) 8 + 9 C 2 (3k) 7 + + 9 C 8 (3k) + 1 =3 ( ) + 1. n 9 3 1. 3 3 mod 3 n = 3k + 1 (k ) 2n =2(3k + 1) 2 1 = 2. n 2 =(3k + 1) 2 1 2 = 1. n 3 =(3k + 1) 3 1 3 = 1. n 9 =(3k + 1) 9 1 9 = 1.
1 2 (53) mod 3 n = 3k 1 (k ) 2n =2(3k 1) 2 ( 1) = 2 1. n 2 =(3k 1) 2 ( 1) 2 = 1. n 3 =(3k 1) 3 ( 1) 3 = 1 2. n 9 =(3k 1) 9 ( 1) 9 = 1 2. (54) n n + 1 (55) n(n + 1) (n + 1) n = 1 2 i.e. n n + 1 2 i.e. n n + 1 2 n(n + 1) 2 (n + 11) n = 11 2 i.e. n n + 11 2 i.e. n n + 11 2 n(n + 11) 2 (56) n, n + 1, n + 2 3 1 3 n(n + 1)(n + 2) 3 k k (57) 2 n(n + 1). 3 n(n + 1)(n + 2). 6 = 2 3 2 3 6 n(n + 1)(n + 2). a, b a n b n = ab n 9 3 3! k k! n k N := n(n + 1)(n + 2) (n + k 1) n+k 1C k = N = k! n+k 1 C k (n + k 1) (n + 2)(n + 1)n k! n+k 1 C k () N k! (58) (n + 5) n = 5 : odd n n + 5 2 n(n + 5). 1 n + 5 n + 2, n + 10 n + 1 (mod 3) n, n + 1, n + 2 1 3 n, n + 5, n + 10 1 3 3 n(n + 5)(n + 10). 2 2 3 1 2 6 n(n + 5)(n + 10). (59) n 3 n = n(n 2 1) = n(n + 1)(n 1). n 1, n, n + 1 1 3 3 n 3 n. i.e. n 3 n (mod 3). 2 n(n + 1) 6 n(n + 1)(n 1). i.e. n 3 n (mod 6).
10 (60) n 5 n =n(n 4 1) =n(n 2 1)(n 2 + 1) =(n 1)n(n + 1)(n 2 + 1). (63) 4 n =(1 + 3) n =1 + n C 1 3 + n C 2 3 2 + + n C n 1 3 n 1 + 3 n 1 (mod 3). ( 1 ) 6 (n 1)n(n + 1). 1 4 n 1 =4 n 1 n n n = 5k, 5k ± 1, 5k ± 2 (k ) =(4 1)(4 n 1 + 4 n 2 + 4 n 3 + + 4 + 1) =3 ( ). 3 4 n 1, n = 5k + 1, 5k, 5k 1 n 1, n, n + 1 5 i.e. 4 n 1 (mod 3). (64) mod 3 2 n = {( 1) + 3} n n = 5k ± 2 =( 1) n + n C 1 ( 1) n 1 3 + + 3 n n 2 + 1 = (5k ± 2) 2 + 1 ( 1) n =(5k) 2 ± 20k + 4 + 1 = 5 ( ). { 2 (n : odd), n 2 + 1 5 1 (n : enen). 5 n 5 n. 2 ( 1, 2, 1, 2,... 2 n+2 2 n ) 6(= 2 3) 5 1 2 2 n+2 2 n =(2 2 1)2 n 30 n 5 n. i.e. n 5 n (mod 30). =3 2 n. 3 2 n+2 2 n. (61) mod 6 a = 6k + 1 b = 6l 2 (k, l ) i.e. 2 n+2 2 n (mod 3). 2 1 = 2 a + b =(6k + 1) + (6l 2) 1 + ( 2) = 1 5. 3a 2b =3(6k + 1) 2(6l 2) 3 + 4 = 7 1. 2 1, 2 3, 2 5, 2. 2 2 = 4 1 2 2, 2 4, 2 6, 1. ab =(6k + 1)(6l 2) 1 ( 2) = 2 4. (65) n = 4k (k N) (62) n 3 + 5n =n 3 n + 6n =n(n 1)(n + 1) + 6n. 6 n(n 1)(n + 1), 6 6n 6 n 3 + 5n. 2 n 3 + 5n, 3 n 3 + 5n 2 n =2 4k =16 k =(1 + 15) k =1 + k C 1 15 + k C 2 15 2 + + 15 k 1 (mod 15).
11 (66) mod 17 n = 8k (k N) (69) mod 7 2 n =2 8k =256 k =(1 + 17 15) k =1 + k C 1 (17 15) + k C 2 (17 15) 2 + + (17 15) k 1. 2 n =2 8k =(2 4 ) 2k =16 2k =( 1 + 17) 2k ( 1) 2k = 1. (67) 5 n 2 n =(5 2) ( 5 n 1 + 5 n 2 2 + 5 n 3 2 2 + + 2 n 1) =3 ( ). 3 5 n 2 n. (68) 30 = 2 3 5 2, 3, 5 2 (70) n 3 3n 2 + 5 3n 1 =3 3(n 1)+1 + 5 3(n 1)+2 =3 27 n 1 + 25 125 n 1. 27 = 7 4 + ( 1) 125 = 7 18 + ( 1) 27 n 1 =( 1 + 7 4) n 1 =( 1) n 1 + n 1 C 1 ( 1) n 2 7 4 + + (7 4) n 1 ( 1) n 1. 3 27 n 1 3 ( 1) n 1. 125 n 1 =( 1 + 7 18) n 1 ( 1) n 1. 25 125 n 1 25 ( 1) n 1. 3 3n 2 + 5 3n 1 3 ( 1) n 1 + 25 ( 1) n 1 =28 ( 1) n 1 =7 4 ( 1) n 1. 7 3 3n 2 + 5 3n 1. 2 5 n 3 n 5 n 3 n 2 n = 2 ( ) 2 n. 2 5 n 3 n 2 n. 3 5 n 2 n 5 n 3 n 2 n = 3 ( ) 3 n. 3 5 n 3 n 2 n. n odd 3 n 2 n =( 3) n 2 n ={( 3) 2} { ( 3) n 1 + ( 3) n 2 2 + + 2 n 1} =5 ( ). 5 5 n 3 n 2 n. 30 5 n 3 n 2 n. n = 3k, 3k ± 1 (k ) (3k) 2 =3 3k 2, (3k ± 1) 2 =(3k) 2 ± 2 3k + ( 1) 2 1 (mod 3). 0, 1. 2 (71) n n = 4k, 4k ± 1, 4k + 2 (k ) (4k) 2 =4 4k 2, (4k ± 1) 2 =(4k) 2 ± 2 4k + (±1) 2 1 (mod 4), (4k ± 2) 2 =(4k) 2 ± 4 4k + (±2) 2 0 (mod 4). 0, 1.2,3
12 (72) mod 6 n n 0, ± 1, ± 2, 3 n n 2 3 n 0 ±1 ±2 3 n 2 0 1 4 3. n 3 n = 6k + 3 n 2 =(6k + 3) 2 =(6k) 2 ± 6 6k + 3 2 9 3. n 2 3 n 3 (73) mod 3 3 n n 2 0 or 1. a 2, b 2 a 2 + b 2 a 2 0 0 1 1 b 2 0 1 0 1 a 2 + b 2 0 1 1 2 a 2 + b 2 0 a 2 0 b 2 0, i.e. a 0 b 0 (74) mod 3 mod 7 mod 7 n n = 7k, 7k ± 1, 7k ± 2, 7k ± 3 (k Z) n n 2 7 n 0 ±1 ±2 ±3 n 2 0 1 4 2. ( (7k ± 3) 2 =(7k) 2 ± 6 7k + (±3) 2 9 2. a 2, b 2 7 0, 7, 14,... a 2 0 b 2 0, i.e. a 0 b 0. (75) ) 2k + 1 2k 1 p { 2k + 1 = pa, 2k 1 = pb (a, b Z) 2 = p(a b). p 2. p p = 2 2k + 1 = 2a 2k + 1 2k 1 p 2k + 1 2k 1 GCD (2k + 1, 2k 1) =(2k 1, 2) =1 ( 2k 1 : odd). 2k + 1 2k 1 (76) 9 9. 8x 8 9 9 x. (77) 7C k = 7! k!(7 k)!. 7! = 7 C k k! (7 k)!. k = 1, 2, 3,..., 6 k!, (7 k)!. 7 7 k! (7 k)! 7 7!. 7 7 C k k! (7 k)!. 7 k! (7 k)! 7 7 C k. p p pc k (k = 1, 2, 3,..., p 1)
(78) 8a 8b (mod 9), i.e. 9 8(a b) 8 9 9 a b, i.e. a b (mod 9). (79) 12 = 3 4 3, 4(= 2 2 ) 3 n(n + 1)(n + 2), 4 n(n + 1)(n + 2)(n + 3). 12 n(n + 1)(n + 2)(n + 3). (80) n n = p a q b r c n 2 n 2 = p 2a q 2b r 2c. x, y p, q, r,... x, y x x =p 2a s 2d = ( p a s d )2 x y (81) b, c : odd b+c, b c : even a : even a = 2z, b + c = 2x, b c = 2y 1 z 2 = xy. 2 x, y p x = pk, y = pl b + c = 2pk, b c = 2pl. b = p(k + l), c = p(k l) 13 b, c x, y 2 x, y x = m 2, y = n 2 1 b + c = 2m 2, b c = 2n 2 i.e. b = m 2 + n 2, c = m 2 n 2 a 2 +b 2 = c 2 (a, b, c) (82) 4 5x 5 4 4 x x = 4k 5 4k = 4y i.e. 5k = y. (x, y) =(4k, 5k) (k Z). (83) 4 (84) 4 2 3 1 = 5 4(x 2) 3(y 1) = 0, i.e. 4(x 2) = 3(y 1). 4 3 (x 2, y 1) = (3k, 4k) (k Z). (x, y) =(3k + 2, 4k + 1) (k Z). (85) N N =5m + 2, N = 7n + 1 5m + 2 = 7n + 1. 5m + 1 = 7n. 5 4 + 1 = 7 3 5(m 4) = 7(n 3). 5 7 (m 4, n 3) = (7k, 5k) (k Z). (m, n) = (7k + 4, 5k + 3) (k Z). N N =5(7k + 4) + 2 =35k + 22 (k Z).
14 (86) x, y 37 13 1 1 1 37 =13 2 + 11, 1 13 =11 1 + 2, 2 11 =2 5 + 1. 3 1 =11 2 5 ( 3 ) =11 (13 11) 5 ( 2 ) =11 6 13 5 =(37 13 2) 6 13 5 ( 1 ) =37 6 13 17. i.e. 37 6 + 13 ( 17) = 1. 4 1 (x, y) = (6, 17). 1 (6, 17) 37x + 13y = a 4 a 37 6a + 13 ( 17a) = a 1 (x, y) = (6a, 17a) (87) (x + 2y)(x 2y) = 5. x + 2y 1 5 1 5 x 2y 5 1 5 1 (88) x 3 3 3 3 y 1 1 1 1 (x 2)(y 3) = 6. } ( ) x 2 1, y 3 2 x 2 1 2 3 6 y 3 6 3 2 1 x 3 4 5 8 y 9 6 5 4 } ( ) (89) x 2 (y 1) 2 = 1 + 13. (x + y 1)(x y + 1) = 12. (x + y 1) (x y + 1) = 2(y 1) : even x + y 1, x y + 1 12 x + y 1, x y + 1 x + y 1 2 6 2 6 x y + 1 6 2 6 2 x 4 4 4 4 y 1 3 3 1. (90) x Z x R x 2 = 5 4y 2 0. y 2 5 4. y y = 0, ± 1 } ( ) y = 0 x 2 = 5 y = ±1 x 2 = 1 x = ±1 (x, y) = (±1, ± 1). (91) x Z x R x 2 = 85 y 2 0. y 2 85. y 2 x 2 y 2 0 1 4 9 16 25 36 49 64 81 x 2 85 84 81 76 69 60 49 36 21 4 x 2 (x, y) =(±9, ± 2), (±7, ± 6), (±6, ± 7), (±2, ± 9) ( ).
15 (92) 2x 2 + (x + y) 2 = 11. (x + y) 2 = 11 2x 2. 1 (x + y) 2 0 { 11 2x 2 0, 2 11 2x 2. 3 2 x 2 11 2 x = 0, ± 1, ± 2 x x 11 2x 2 x 0 1 1 2 2 11 2x 2 11 9 9 3 3. ( 3 ) (93) ab ac bc abc =bc + ca + ab bc + bc + bc =3bc. bc > 0 (1 )a 3. a = 1, 2, 3 a = 1 bc = bc + b + c i.e. b + c = 0. a = 2 2bc = bc + 2c + 2b. bc 2b 2c = 0. (b 2)(c 2) = 4. 1 : y = x ± 11 2x 2. x = 1, y = 1 ± 3 = 2, 4 x = 1, y = 1 ± 3 = 4, 2. 4 2 = a b c 0 b 2 c 2 (b 2, c 2) = (1, 4), (2, 2), i.e. (b, c) = (3, 6), (4, 4) a = 3 (x, y) =(1, 2), (1, 4), ( 1, 4), ( 1, 2). ( y 4 ) y 2 y 2 + 2x y + (3x 2 11) = 0. y = x ± 11 2x 2. ( ) 4 0 y y 2 ( ) /4 ( ) 4 /4 0 x 3bc = bc + 3c + 3b. bc 3 2 b 3 2 c = 0. ( b 3 ) ( c 3 ) = 9 2 2 4. (2b 3)(2c 3) = 9. 3 = a b c 3 2b 3 2c 3 (2b 3, 2c 3) = (3, 3), i.e. (b, c) = (3, 3) (a, b, c) = (2, 3, 6), (2, 4, 4), (3, 3, 3). abc 1 a + 1 b + 1 c = 1
16 (94) a b c abc =a + b + c c + c + c =3c. c > 0 (1 )ab 3. ab = 1, 2, 3 ab = 1 a, b = (1, 1) c = 1 + 1 + c. ab = 2 (a, b) = (1, 2) 2c = 1 + 2 + c. i.e. c = 3. c b = 2 ab = 3 (a, b) = (1, 3) 3c = 1 + 3 + c. i.e. c = 2. c b (a, b, c) = (1, 2, 3) (97) 2 = 2 i 2 j. 1 = 2 i 1 2 j 1. 2 1 odd 2 i 1 even ( i 1 1) 2 j 1 odd. j 1 = 0 i.e. j = 1. 2 1 = 2 i 1 1. 2 i 1 = 2. i 1 = 1, i.e. i = 2. 1 m = 3 (m, n) = (3, 3). 2n + 6 n 2 N 2n + 6 + n n 2 + n 1 n 2 + n > 0 2n + 6 n 2 + n. n 2 n 6 0. (n + 2)(n 3) 0. n + 2 > 0 (95) 2000 = 2 (2 5) 3 = 2 4 5 3 2 x (2y + 1) = 2 4 5 3. 2y + 1 2 x = 2 4. 2y + 1 = 5 3. x = 4, y = 62. (96) 2 n = (m + 1)(m 1). (m + 1) (m 1) = 2 : even m + 1, m 1 m+1, m 1 { m + 1 = 2 i, m 1 = 2 j (1 j < i) 1 n 3 0. n = 1, 2, 3. f(n) = 2n + 6 n 2 + n f(1) = 8 2 = 4 N, f(2) = 10 / N, 6 f(3) = 12 12 = 1 N. f(n) N n( N) n = 1, 3 (98) n 2 + 3 = m (m N) n 2 + 3 = m 2. m 2 n 2 = 3. (m + n)(m n) = 3. m + n > m n > 0 (m + n, m n) = (3, 1). m = 2, n = 1.
(99) N =1101 (2) =1 2 3 + 1 2 2 + 0 2 1 + 1 1 }{{} 2 0 =8 + 4 + 1 = 13. (100) 1 ( ) 2 n 0 1 2 3 4 5 6 2 n 1 2 4 8 16 32 64 N =100 =64 + 36 =64 + 32 + 4 =1 2 6 + 1 2 5 + 1 2 2 =1100100 (2). 2 ( ) N 0 = + a 3 2 3 + a 2 2 2 + a 1 2 1 + a 0 1 =2 ( + a 3 2 2 + a 2 2 1 + a ) 1 1 +a 0 }{{} N 1 a 0 N 0 2. N 1 := N 0 a 0 2 = + a 3 2 2 + a 2 2 1 + a 1 1 =2 ( + a 3 2 1 + a 2 1 ) }{{} N 2 +a 1 a 1 N 1 2. ( ) N 0 := 100 = 2 50 + 0, a 0 = 0, N 1 = 50. N 1 = 50 = 2 25 + 0, a 1 = 0, N 2 = 25. N 2 = 25 = 2 12 + 1, a 2 = 1, N 3 = 12. ( ) N 3 = 12 = 2 6 + 0, a 3 = 0, N 4 = 6. ( ) N 0, N 1, N 2, 2 17 2 ) 100 2 ) 50 0 = a 0 2 ) 25 0 = a 1 2 ) 12 1 = a 2 2 ) 6 0 = a 3 2 ) 3 0 = a 4 }{{} 1 1 = a 5 a 6 (101) N 10 N =454 (6) =4 6 2 + 5 6 1 + 4 1 =4 36 + 5 6 + 4 =144 + 30 + 4 = 178. 2 1 3 2 n 0 1 2 3 4 3 n 1 3 9 27 81 2 3 n 2 6 18 54 162 N =178 2 =162 + 16 =162 + 9 + 7 =162 + 9 + 6 + 1 =2 3 4 + 1 3 2 + 2 3 1 + 1 1 =20121 (3). 178 2 1 3 ) 178 3 ) 59 1 = a 0 3 ) 19 2 = a 1 3 ) 6 1 = a 2 }{{} 2 0 = a 3 a 4 N 4 = 6 = 2 3 + 0, a 4 = 0, N 5 = 3. (102) M + 1 = 666 }{{ 66 } (7) + 1 N 5 = 3 = 2 1 + 1, a 5 = 1, N 6 = 1(= a 6 ). 6 n =6 1 + 6 7 1 + 6 7 2 + + 6 7 n 1 + 1 N = 1100100 (2). =6 7n 1 7 1 + 1 = 7n.
18 666 }{{ 66 } (7) + 1 6 n =1 000 }{{ 00 } (7). 0 n (103) ( ) A = B A = B ( ) 2 1 ( ) a n > b n A B =(a n b n ) 7 n + (a n 1 b n 1 ) 7 n 1 + + (a 1 b 1 ) 7 1 + (a 0 b 0 ) 1 1 7 n 6 7 n 1 6 7 1 6 1 =7 n 6 7n 1 7 1 =1 > 0. A > B a n < b n (105) P (n) : a a n = b n n a n+1 n = 1, 2, 3, a n 1 a 1 a 0(7) = b n 1 b 1 b 0(7) 1 P (1) : a 1 a 2 1 a n 1 = b n 1 ( ) 2 n P (n) P (n+1) : a n+1 a n+2 2 ( ) P (n+1) A = B 7 (a n 7 n 1 + a n 1 7 n 2 + + a 1 1 ) + a 0 =7 (b n 7 n 1 + b n 1 7 n 2 + + b 1 1 ) + b 0 7 a 0 = b 0. a n 7 n 1 + a n 1 7 n 2 + + a 1 =b n 7 n 1 + b n 1 7 n 2 + + b 1 a 1 = b 1 ( ) A =a n a n 1 a 1 a 0(7) =a n 7 n + a n 1 7 n 1 + + a 1 7 1 + a 0 1 (a 0, a 1,, a n 1, a n ) 7 n A min A = 000 }{{ 00 } (7) = 0 0 n max A = 666 }{{ 66 } (7) = 6 7n 1 7 1 = 7n 1 6 n 7 n 7 n A min A = 0 max A = 7 n 1 (a 0, a 1,, a n 1, a n ) 7 (2 10 ) a n+1 a n+2 p a n+1 = pi, a n+2 = pj 2 a n =a n+2 5a n+1 = p (j 5i). a n a n+1 p P (n) P (n + 1) 1, 2 P (1), P (2), P (3), 2 : a n+2 = a n+1 5 + a n (a n+2, a n+1 ) = (a n+1, a n ) (n = 12, 3, ). n 7 n (a n+1, a n ) = (a 2, a 1 ) = 1 i.e. a n+1 a n
19 (106) 2 2 2 = p q 2q = p. 2q 2 = p 2. (p, q N) p 2, q 2 2 {,. 2 (107) b 0 2 = a b. { / Q, Q b = 0. a = 0. (108) (109) (a a ) + (b b ) 3 = 0. p q a a = 0, b b = 0. i.e. a = a, b = b. ( ) 1 ( ) 3 ( ) 2 p p a + b + c q q ( p q ap 3 + bp 2 q + cpq 2 + dq 3 = 0. 3 p p (ap 2 + bpq + cq 2 ) = q 3 d. p q 3 p d. ) + d = 0. 1 3 q q (bp 2 + cpq + dq 2 ) = p 3 a. q a. 1 ( ) d a (3 )