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2012 3 17
在數學競賽中,經常會問到有關最大公因數(H.C.F.)和最小公倍數(L.C.M.)的問題。
(numbers)
(factor) (Highest Common
Factor) (Least Common
Multiple)
12 1 2 3 4 6 12
18 1 2 3 6 9 18
1 2 3 6 6
4 4 8 12 16 20 24 …
6 6 12 18 24 30 36 …
12 24 12
1
(relatively prime)
12 322 ×× 18 332 ××(Product of Prime Factors)
(unique) 632 =×363322 =××× 12 18
48 42
48 42
48 42 2 2
方法 。
2 48 42
3 24 21
8 7
48 42 2 2
48 42
632 =× 48 42 3368732 =×××
6
56
2
H.C.F. L.C.M. 18 24 32 H.C.F.
L.C.M.
2 18 24 32
9 12 16
9 12 16 H.C.F. 2 L.C.M. 3456161292 =×××L.C.M.
9 12 12 16
2 18 24 32
2 9 12 16
2 9 6 8
3 9 3 4
3 1 4 ←三數再無公因子
18 24 32 L.C.M. 2884133222 =××××××
32 24 18
288 932288 =÷ 1224288 =÷1618288 =÷ 172816129 =××
288
2884133222 =××××××(index notation)
422222 =××× 2 (base) 4 (index) 2
25 32288 ×=
1
5766 31
5766 2 3 312= × × 31
31 2 3 31 186× × =
2 31 62× = 3 31 93× =
3
1186 1 1661 2 2209 3
118511186 =−
165921161 =−
221232209 =+
1185 1659 2212
1185 3 5 79= × ×
1659 3 7 79= × ×
2212 2 2 7 79= × × × 79
165 15
165
165 3 5 11= × ×
= ×15 11 15 15x 15y
x + y = 11
x = 1 y = 10 15 150
x = 2 y = 9 30 135
x = 3 y = 8 45 120
x = 4 y = 7 60 105
x = 5 y = 6 75 90
x > 6
100 6666
100 100
66.661006666 =÷ 6666
Q 1011006666 ×=
99 66 1 132266 =× 100
66
14 30 33 35 39 75 143 169
4
8
14 = 2 × 7
30 = 2 × 3 × 5
33 = 3 × 11
35 = 5 × 7
39 = 3 × 13
75 = 3 × 25
143 = 11 × 13
169 = 213
↑ ↑ ↑
↑
{14} {30, 35} 11 33 143
A B
14, 33 30, 35, 143
14, 143 30, 35, 33
B 143 13 39 B 169
A
A B
14, 33, 75, 169 30, 35, 143, 39
14, 143, 39, 75 30, 35, 33, 169
5
3 2 5 3 7 2 (solutions) 23 23 105
3 70 5 21 7 15 3
105 105
2 70 3 21 2 15 233× + × + × = 233 105 233 105 128 105 23
23 128 233 338……
3 70
5 21
7 15
3 105 3 5 7
5 2 7 3 11 4
7 3 11 4 59 7 11
77 59 77 5 2 59
59 + 77 = 136 5 1
136 + 77 = 213 5 3
213 + 77 = 290 5 0
290 + 77 = 367 5 2
367
7 3 11 4
6
2 3 4 5 6 1 61
2 3 4 5 6 60 61 60 7
61 7
61 + 60 = 121 7 2
121 + 60 = 181 7 6
181 + 60 = 241 7 3
241 + 60 = 301 7 0
301
6 7 8 9
6 7 8 9 504 504 10000
6 7 8 9 10000 504 19÷ = ......424 504
957619504 =×
504 10080
99999 504 99999 504 198÷ = ......207
504 99792198504 =×
10080 504 20
99792 504 198
10080 99792 198 – 20 + 1 = 179
2C(94)
136
73
9
2
xm
m
x
x
9
2
136
73 =−−
m
(73 – x) 2 (136 – x) 9 m
(136 – x) – (73 – x) = 9m – 2m
7m = 63 m = 9
∴ 9273 ×=− x 99136 ×=− x
x = 55
9m(73 – x) = 2m(136 – x)
m[9(73 – x) – 2(136 – x)] = 0 Q m 0
∴ 9(73 – x) – 2(136 – x) = 0
7x = 385
x = 55
8
1. 320 240 200
2. 52 57 65 69 95 28 9 161 3. 21 22 34 39 44 45 65 76 133 153
4. 1111 1
5. 433 260 30 50
13 8 6. 115 142 72 7 2
7. 360 314 245 8. 9cm 6cm 7cm 9. 3 4 5 6 2 7 10. 5 2 7 3 9 4 11. 10 9 8 7 6 5
4 3 2 1 12. 5 6 7 1 10000
13. 6
10
14. 20
15
25
9
1. 320 240 200
52320 6 ×=
532240 4 ××=
23 52200 ×=
40523 =×
2. 4
9 = 23
28 = 22 × 7
52 = 22 × 13
57 = 3 × 19
65 = 5 × 13
69 = 3 × 23
95 = 5 × 19
161 = 7 × 23
2 2
( ) 28 52
( ) ( ) {28,69} {52,161}
( ) {28, 57, 69} {9, 52, 161}
( ) {28, 57, 65, 69} {9, 52, 95, 161}
3. 2
21 = 3 × 7
22 = 2 × 11
34 = 2 × 17
39 = 3 × 13
44 = 22 × 11
45 = 23 × 5
65 = 5 × 13
76 = 22 × 19
133 = 7 × 19
153 = 23 × 17
10
19
( ) 76 133
( ) {21, 76} {133}
( ) 21 39 3 39 133 {21, 76}
{39, 133}
( ) ( ) {21, 65, 76} {39, 45, 133}
( ) 45 23 45 153 ( )
153 17 34 153 {21, 65,
76, 153} {34, 39, 45, 133}
( ) ( ) {21, 22, 65, 76, 153} {34, 39, 44, 45, 133}
4. 1111 1111
101111111 ×=
101×a 101×b 101×c 101×d 11=+++ dcba
101
5. 433 13 260 8 420 252
7532420 2 ×××=
732252 22 ××=
847322 =×× 30
50 42732 =××
6. 115 142 72 7 2
115 – 7 = 108
144 + 2 = 144
108 144 72 = 36
7. 360 314 245
360 – 314 = 46 = 232×
314 – 245 = 69 = 233×
360 – 245 = 115 = 235×
23
8. 9cm 6cm 7cm
9 6 7 9 6
18 7 9 6 (relatively prime)
126187 =×
= )7126()6126()9126( ÷+÷+÷ = 5292
11
9. 3 4 5 6 = 605322 =×× 62
62 60 7
62 + 60 = 122 7 3
122 + 60 = 182 7 0
182
10. 5 2 7 3 17 5 7 35 17
35 9 4
17 + 35 = 52 9 7
52 + 35 = 87 9 6
87 + 35 = 122 9 5
122 + 35 = 157 9 4
157 + 35 = 192 9 3
192
11. 10 9 … 2 = 2520
5000 10 9 … 2 1
= 122520 −×
= 5039
12. 5 6 7 L.C.M. = 765 ×× = 210
5 6 7 1 = 210 + 1 = 211
130......4721010000 =÷
5 6 7 1 = 147210 +×
= 9871 < 10000
47
13. 6 10
6 10
Q 6 10 = 30
∴ = 630÷
= 5
= 1030÷
= 3
= 5 – 3
= 2
∴ 15230 =÷