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- しおり おおばま
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1 I
2 1 1.0 ( ) ( ) 1 2 2
3 ( ) 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
4 ? ( ) ( ) + ( ) ( ) = 14, ( ) = 26, ( ) = 10 ( ) ( ) + ( ) = = 374 ( ) = ( ) ( ) + ( ) ( ) 1 ( ) = ( ) ( ) + ( ) (1) 34 3 (2) (3) ( ) a b q r 4 a = bq + r ( 0 < = r < b) 1 4 q r quotient remainder 4
5 1.2 0 ( ) a b 0 a = bq a b b a b a a b ( ) 24 4 (!) 6 24 = ( ) a a = 1 a 1 0 = 0 a = 2 3 5
6 2 6 6 = = = ( 2 ) ( 3) ?! 6 = , 2, 3, 6? , 2, 3, 6 ( ) = = = 4 4 6
7 a a > = 4 b 16 = a b b a ( ) 2 (1) 12 (2) 28 (3) 60 (4) 29 a a = a 1 1 a a ? (4)
8 ( ) ( ) 1 1 ( 1) ? ? ( ) b > 4 a b > 16! a b = 16!! 6 8
9 20 7? 2( (629 )) = ( ) 1 ( ) 8 ( ) = = a = p q p a q a 9
10 ( ) = = = ( ) 2 ) ) ) ) 63 3 ) =
11 2 504 = ( ) 3 (1) 45 (2) 60 (3) 180 (4) 1024 (5) ( ) = 2 3 = = = ( ) 11
12 12 12 = = = ( ) ( ) a b a b a b b = aq ( ) a q b = a q b a b ( )
13 ( ) a b a b 12 a b b a b = a a = a = a = q b = aq a b ( ) 504 = , 36 = = ( ) (2 7) 504 = ( ) (1) 12 (2) 34 (3) 63 (4) 72 (5) A B B A A B A B B A A B B A 13
14 (0 ) a( 0) a 0 = 1 ( ) = 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) , , , , , , , , ,
15 1, 2, 4, 8 3, 6, 12, 24 9, 18, 36, , 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 ( ) ( ) 5 (1) 20 (2) 63 (3) 256 (4) = 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 , 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300 ( )
16 ( ) (2 ) (5 ) a + 2b b a + 2b = 2(5a + b) 2 ( ) 2 ( ) a 2b 346 a = 34, b = 3 a b 16
17 (4 ) a + b ( b ) 4 b = 4k (k ) 100a + b = 4(25a + k) 4 ( ) ( ) 7 4 (1) 172 (2) 122 (3) 2348 (4) (5) (8 ) ( ) 8 8 (1) 172 (2) 122 (3) 2348 (4) (5)
18 (3 9 ) a, b, c, d 1000a + 100b + 10c + d 15 9 a + b + c + d a + 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) a + 100b + 10c + d 9 ( ) = = ( ) ( a = 2, b = 3, c = 4, d = = 10 3, 100 = 10 2, 10 = 10 1, 1 = a b c d p n, 9m 9n + 9m 18
19 ( ) a a = 2k 2 3 k 3 k = 3l a = 2 (3l) = 6l a 6 ( ) , 136, 275, 635, 1080, (1) 3 (2) 6 (3) 9 (4)
20 1.6.1 ( ) 2 ( ) 12 1, 2, 3, 4, 6, , 2, 3, 6, 9, , 2, 3, 6 ( ) 10 (1) 24, 40 (2) 12, 36 (3) 40, 72, 96 (4) 14, 33 1, 2, 3, 6 6 ( ) ( ) 8 ( ) 20
21 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 = =
22 2, 2, = 12 ( ) 2 ) 72, 84 2 ) 36, 42 3 ) 18, 21 6, 7 2, 2, 3 48, 36, = 6 2 ) 36, 48, 90 3 ) 18, 24, 45 6, 8, 15 ( ) 11 (1) 24, 40 (2) 12, 36 (3) 40, 72,
23 ( ) 1 ( ) ( ) 12 (1) 6, 8 (2) 15, 49 (3) ( ) 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 ( ) (!) = = 2 3 ( ) 13 (1) 24 (2) 12 (3)
24 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
25 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 ( ) = = = = ( ) ( ) 14 (1) 56, 432 (2) 16632, ( ) 2 ( ) 25
26 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
27 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 = = p, q 6, = = = ( ) 2, 2, 3 6, 7 27
28 2 ) 72, 84 2 ) 36, 42 3 ) 18, 21 6, 7 16 (1) 9, 12 (2) 24, 36 (3) 32, 80 36, 48, = = = = 720 ( ) 11 2 ) 48, 36, 90 3 ) 24, 18, 45 2 ) 8, 6, 15 3 ) 4, 3, 15 4, 1,
29 17 (1) 12, 18, 24 (2) 7, 13, 15 (3) 28, 42, 70 (4) 126, 196, (!) ( 12 18) OK ( 20 ) a, b d l ab = dl 12 3 a, b a b 20 29
30 21 a = 21p, b = 21q p q ab = p 21q = pq = = 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 = p 21q = 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 = a = 21 26, b = 21 a = 42 b = 273 a = 546 b = 21 ( ) ( ) 30
31 1.7 ( ( ) n > = 3 x n + y n = z n 31
32 (x, y, z)
33
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数学のかたち 共線問題と共点問題 Masashi Sanae 1 テーマ メネラウスの定理 チェバの定理から 共線問題と共点問題について考える 共線 点が同一直線上に存在 共点 直線が 1 点で交わる 2 内容 I. メネラウスの定理 1. メネラウスの定理とその証明 2. メネラウスの定理の応用 II. 3. チェバの定理とその証明 メネラウスの定理 チェバの定理の逆 1. メネラウスの定理の逆
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