แผนการจัดการเรียนรู้ที่ 1 ตัวหารร่วมมากและตัวคุณร่วมน้อย

1
1
1
1
14
1
K-P-A
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1.
2.
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1.
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1
2.
3.

4.
8 1 2 4 8
1. 20
1.1 ,
1.2
1.3 20 1 2 4,5 10,20
3.
8
16
24
32
40

4. “ ”
5.
6. 1.1
7.
8. 1.1
9.
18 39
10. –
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1.
1
2.
3. 1 1
4. 1.1
K-P-A

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K
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5
P
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A
1
2.
3.
4.
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2

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1
6
10
36
1 2 3 6
1 2 5 10
1 2 3 4 6 9 12 18 36
1 2 3 6 6 6
1 2 5 10 10 10
1 2 3 4 6 9 12 18 36 36
36
2 12
1 12
1
12 = 12
2 12
2
12 = 6
3 12
3
12 = 4
4 12
4
12 = 3
6 12
6
12 = 2
12 12
12
12 = 1
1 2 3 4 6 12 12

1.1
1 1
1.
16
22
46
72
1 2 4 8 16
1 2 11 22
1 2 23 46
1 2 3 4 6 8 9 12 18 24
36 72
2.
87 = 1 3 29 87
(1) 7 = 1 7
(2) 9 = 1 3 9

(3) 24 = 1 2 3 4 6 8 12 24
(4) 35 = 1 5 7 35
(5) 100 = 1 2 4 10 25 50 100
1.1
1 1
1. 5
16
22
46
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72 __________________________________________
2. 5
87 = 1 3 29 87
1 7 = ______________________
2 9 = ______________________
3 24 =
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4 35 =
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5 100 =
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2
1
1
1
14
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2
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2
2 1
K-P-A
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1.
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2
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26  -
29 - 
50  -
1.
1.1
2 2
1.2
1

1.3 2 4 8 12
16 …
1.4 2
2. 10
19 23 36 404 523
724 396 1 001 1 236 3 993
3.
5. 2.1
6.
7.
8.
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9. –
1.
1
2. 2.1

3. 2.1
2.1
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16 1 2 4 8 16 2
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6
2
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12  -
15 - 
18  -
24  -
27 - 
42  -

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2.1
2 1
1.  “ ” “ ”
1  -
2 - 
3 1 10 13 17 - 
4 8 36 44  -
5 2
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2.
1

2
3
3.
17 18 19 20 21 22 23 24
18 22,24
17 19 21 23
2.1
2 1
1.  “ ” “ ”
5

1
2
3 1 10 13 17
4 8 36 44
5 2
“ ”
2. 3
1
______________________
2
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3
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3. 2
21 28 33 67 84 102 113 149
______________________
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3
1
1
1
14
1
1
1
K-P-A
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1.
2.

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1.
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4.
1
1
2,3,4,5,6,7
2 1
3 1
4
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5 1
6 1 2 3
7 1
2,3,5,
1

1.
1.1
1.2
2 2 3 5 7
1.3 2
2.
2
3. “ ”
4. 3 – 4 2
5.
1
1
1 1
“ ”

“ ”
6.
7. 3.1
7. 3.1
8. 1 2 3
9.
10. –
1.
1
2. 3.1
3. 3.1
3.1
K-P-A
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K
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5

P
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2
A
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2

3
1
1
1
6 11 27 41 67 91 103
6 1 2 3
11 1
27
1 3 9
41 1
3.1

3.1
67 1
91 1 7 13
103 1
11 41 67 103
1
1

3 1
1. 1
100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

3.1
3 1
1. 1
100 10
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100
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4
1
1
1
14
1
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K-P-A
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1.
2.
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1
2
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4.
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20
26
32
38
42
1 2 4 5 10
20
1 2 13 26
1 2 4 8 16
32
1 2 19 38
1 2 3 6 7
14 42
2 5
2 13
2
2 19
2 3 7

1.
2.
-
-
3. 4.1
4.
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5.
6. 4.1
7.
8. -
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1.
1
2. 4.1
3. 4.1
4.1
K-P-A
K
-
4.1
- -
5
P
-
-
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4.1
-
2

4.1
4 1
“ ”
20
26
32
38
42
1 2 4 5 10
20
1 2 13 26
1 2 4 8 16
32
1 2 19 38
1 2 3 6 7
14 42
2 5
2 13
2
2 19
2 3 7

4.1
4 1
1.
17
37
46
1 17
1 37
1 2
17
37
2 46

51
67
23 46
1 3
17 51
1 67
3 17
67
2.
1 18 = 2 3
2 27 = 3
3 35 = 5 7
4 40 = 2 5
5 70 = 2 5 7

4.1
4 1
1. 5
17
37
46
51
67
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2. 5
1 18 = ______________________

2 27 = ______________________
3 35 = ______________________
4 40 = ______________________
5 70 = ______________________
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5
1
1
1
14
1

K-P-A
:
1.
2.
:
1.
:
4.
12 = 2 2 3
18 = 2 3 3
1
1 120
120

2 60
2 3 20
2 3 2 10
2 3 2 2 5
120 = 2 2 2 3 5
2
2 360
2 ) 240
2 ) 120
2 ) 60
3 ) 30
3 ) 15
5
360 = 2 2 2 2 3 5
1.
-
2
-
1 120
120

2 60
2 2 30
2 2 2 15
2 2 2 3 5
240 = 2 2 2 3 5
2 180
2 ) 180
2 ) 90
3 ) 45
3 ) 15
5
180 = 2 2 3 3 5
3.
4. 5.1
5.
6. 5.1
7.
8. -

1.
1
2. 5.1
3. 5.1
5.1
K-P-A
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5.1
- -
5

P
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5.1
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2
A
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2

5 1
12 = 2 2 3
18 = 2 3 3
1
1 240
5.1

240
2 120
2 2 60
2 2 2 30
2 2 2 2 15
2 2 2 2 3 5
240 = 2 2 2 2 3
5
2
2 360
2 ) 360
2 ) 180
2 ) 90
3 ) 45
3 ) 15
5
360 = 2 2 2 3 3 5
5.1
5 1
1.
1 175

175
5 35
5 5 7
175 = 5 5 7
2 196
196
2 98
2 2 49
2 2 7 7
196 = 2 2 7 7
2.
1 180
2 ) 180
2 ) 90
3 ) 45
3 ) 15
5
180 = 2 2 3 3 5
2 425
5 ) 425
5 ) 85
17
425 = 5 5 7

5.1
5 1
1. 5
1 175
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2. 5
1 180
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6
1
1
1
14
1
4

1. . . .
1 . . . 36 60
36 1,2, 3,4, 6,9,12,18 36
60 1, 2, 3,4, 5,6,
10,12,15,20,30 60
36 60 1,2,3,4,6,12
36 60 12
. . . 36 60 12
12
2. . . .
2 . . . 24 36
24 36
24= 2 2 2 3
36= 2 2 3 3
24 36 2 2 3
24 36 2 2
3 = 12
. . . 24 36 12

12
1.
2. 2
18 1, 2, 3, 6, 9 18
24 1, 2, 3, 4, 6, 8, 12 24
3.
18 24
18 24 1, 2, 3, 6
6 6 ( . . .)
18 24
4.
. . .
4.1 . . .
1 . . . 16 24
16 1, 2, 4, 8 16
24 1, 2, 3, 4, 6, 8, 12 24
16 24 1, 2, 4 8

16 24 8
. . . 16 24 8
8
4.2 . . .
2 . . . 24 30
24 30
24 = 2 2 2 3
30 = 2 3 5
24 30 2, 3 2 3
24 30 2 3
= 6
. . . 24 30 6
6
5. 2 3
2-3
6. 6.1
7. . .
8. . .
. . 2

9. 6.1
10.
11. -
1.
1
2. 6.1
3. 6.1
4. . .
6.1
K-P-A
-
K
-
6.1
- -
5

P
-
-
-
- -
6.1
-
2
A
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2
*
*

6 1
1. . . .
1 . . . 15 20
15 1, 3, 5, 15
20 1, 2, 4, 5, 10 20
15 20 1, 5
15 20 5
. . . 15 20 5
5
2. . . .
6.1

2 . . . 16 24
16 24
16 = 2 2 2 2
24 = 2 2 2 3
16 24 2 2 2
16 24 2
2 2 = 8
. . . 16 24 8
8
6.1
6 1
1.

20 45
20 1, 2, 4, 5, 10 20
45 1, 3, 5, 9, 15 45
20 45 1, 5
20 45 5
. . . 20 45 5
5
2 25 60
25 1, 5 25
60 1, 2, 3, 4, 5 6 10, 12,
15, 20 30 60
25 60 1, 5
25 60 5
. . . 25 60 5
5
2
= 3 7
105 = 3 5 7
= 21

6.1
6 1
1.
20 45 2
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2 25 60 2
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2
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K-P-A
:
:
1.
:
4.
. . .
. . . 27, 36 45
3 ) 27 36 45
3 ) 9 12 15
3 4 5

. . . 27, 36 45 3 3 = 9
9
1. . .
2. 3-4
3.
1 . . . 12, 24 60
2 ) 12 24 60
2 ) 6 12 30
3 ) 3 6 15
1 3 5
. . . 12, 24 60 2 2 3 = 12
12
2 . . . 12, 18 24
2 ) 12 18 24
3 ) 6 9 12
2 3 4
. . . 12, 24 60 2 3 = 6
6

4.
4.1 3
4.2 3
4.3
5.
2-3
6. . .
7. 5
7.1 . . .
7.1 . . .
8. . .
9. -

1.
1
2. 7.1 . . .
3. 7.1 . . .
7.1 . . .
K-P-A
-
K
-
7.1
. . .
- -
5

P
-
-
-
- -
7.1
. . .
-
2
A
-
-
-
-
-
2

1 . . . 27, 36 45
3 ) 27 36 45
3 ) 9 12 15
3 4 5
. . . 27, 36 45 3 3 = 9
9
2 . . . 12, 18 24
2 ) 12 18 24
3 ) 6 9 12
2 3 4
. . . 12, 24 60 2 3 = 6
6

7.1 . . .
7 1
1
2 ) 28 80
2 ) 14 40
7 20
=
3 ) 36 54 63
3 ) 12 18 21
2 ) 4 6 7
2 3 7
=
3 18 78
18 78

2) 18 54 78
3 ) 9 27 39
3 9 13
15 78 2 =6
6
7.1 . . .
7 1
1.
2
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4
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3 15 78 4
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8
1
1
1
14
1
4

K-P-A
:
:
1.
:
4.
45 126
1 45 126 2
36 45
9 36 4
36
0
1 126
45 = 2 36
2 1 36
45 36 =1 9

3 2 9 36
36 9 = 4 0
45 126 9
1. 1 3
- . .
-
-
2. . .
3.
2
1 210 385
1 210 385 1
175 210
35 175 5
175
0
1 385 210 =
1 175

2 1 175 210
175 = 1 35
3 2 35 175
175 35 = 5 0
210 385 35
2 78 90
6 78 90 1
72 78
6 12 2
12
0
78 90 6
4. 3 - 5
5. . .
6. 8.1
8.1
7. 1 . .

8. -
1.
1
2. 8.1
3. 8.1
8.1
K-P-A
-
K
- 8.1
-
P
-
-
-
- - 8.1
A
-

8 1
45 126
1 45 126 2
36 90
9 36 4
36
0
1 126
45 = 2 36
2 1 36
45 36 =1 9
3 2 9 36
36 9 = 4 0
45 126 9
8.1

8.1
2 78 90
6 78 90 1
72 78
6 12 2
12
0
78 90 6
8 1
1
1 85 136
1 85 136 1
51 85
34 51 1

34 34
17
85 136 17
2 105 195
1 105 195 1
90 85
15 90 6
90
0
105 195 15
3 255 795
8 255 795 3
240 765
15 30 2
30
0
255 795 15
8.1

8 1
1
1 85 136 2
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2 105 195 4
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3 255 795 4
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9
1
1
1
14
1
4
K-P-A
:

. . .
:
1.
:
4.
1 15 23 31
1 2 3
15 1 15-1
14
23 2 23-2
21
31 3 31-3
28
14 21 28
14 21 28
. . . 14 21 28
7 ) 14 21 28
2 3 4
. . . 14 21 28 7

15 23 31
1 2 3 7
7
2
108
84
. . .
84 108
. . . 84 108
3 ) 84 108
4 ) 28 96
7 24
. . . 84 108 3 4 = 12
108
84
12108
1284 =
9
7
9
7
1. . . .
2. . . .

2 - 3
1 1,504
1,697 1
1,697 1
(1,697 - 1) 1,696
1,504
1,697 1 . . . 1,504 1,696
. . . 1,504 1,696
2 ) 1,504 1,696
2 ) 752 848
2 ) 376 424
2 ) 188 212
2 ) 94 106
47 53
. . . 1,504 1,696 2 2 2 2 2
= 32
1,504
1,697
1 32
32
2
875,1
315

. . .
315 1,875
. . . 315 1,875
3 ) 3151,875
5 ) 105 625
21 125
. . . 315 1,875 3 5 = 15
875,1
315
15875,1
15315
=
125
21
125
21
3. 9.1
4. 3 – 4
“ 63 90
”
. . . 63 90
63 = 3 3 7
90 = 3 3 2 5

. . . 63 90 3 3 =
9
9
63 9 = 7
90 9 = 10
9
9
7 10 = 70
9
70
5.
. .
6. 9.1

7. . .
8. -
1.
1
2. 9.1
3. 9.1
9.1
K-P-A
-
K
-
9.1
- -
5
P
-
-
- -
9.1
-
2

9 1
1 15 23 31
1 2 3
15 1 15-1
14
23 2 23-2
21
31 3 31-3
28
14 21 28
14 21 28
. . . 14 21 28
9.1

7 ) 14 21 28
2 3 4
. . . 14 21 28 7
15 23 31
1 2 3 7
7
2
108
84
. . . 84 108
. . . 84 108
3 ) 84 108
4 ) 28 96
7 24
. . . 84 108 3 4 = 12
108
84
12108
1284 =
9
7
9
7
9.1
9 1

1.
252
108
. . . 108 252
3 ) 108 252
4) 36 84
3) 9 21
3 7
. . . 108 252 3 4 3 = 36
252
108
7
3
36252
36108
7
3
160 80
70
2 ) 160 80 70
5) 80 40 35
16 8 7
160 80 70 2 5 =
10
10
10
160 = 16
10
80 = 8
10
70 = 7

16 + 8 + 7 = 31
31
9.1
9 1
1.
252
108 5
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160 80
70
5
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10
1
1
1
14
1
3

K-P-A
:
1.
2.
3.
:
1.
:
4.
2, 4, 6, 8, 10, 12, 14, 16, 18,…
2
3, 6, 9, 12, 15, 18,… 3
6, 12, 18,... 2 3
6,12,18,... 2 3
2 3

. . . 2 3 . . . 2 3
6
. . .
. . .
1
. . . 8 12
8, 16, 24, 32, 40, 48,…
8
12, 24, 36, 48,… 12
24, 48, ... 8 12
24 8 12
. . . 8 12 24
24
2
. . . 6, 20 30
6, 20 30
6 = 2 3
20 = 2 2 5
30 = 2 3 5
6, 20 30
6, 20 30 2 2
3 5 = 60
. . . 6, 20 30 60

60
1. . .
. .
2. 2 2, 4, 6,
8, 10, ... 2, 4, 6, 8, 10, ... 2
3, 4, 5
2, 4, 6, 8, 10, 12, 14, 16, 18, ...
2
3, 6, 9, 12, 15, 18, 21, 24, 27 ...
3
4, 8, 12, 16, 20, 24, 28, 32, ... 4
3. 2
3 6, 12, 18, ... 2 3 6, 12, 18,
... 2 3 2 3
( . . .) . . . 2 3 6
. . . 3 4 2 4
4. . . .
2 . . . 2 - 3

1 . . . 6 15
6 15
6 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
15 15, 30, 45, 60, ...
6 15 30, 60, ...
6 15 30
. . . 6 15 30
30
5. . . .
5.1 . . . 4.4
. . .
. . . 6 15
6 = 2 3
15 = 3 5
. . . 6 15 6 15
6 15
. . . 6 15 3 2 5 =
30
5.2 2 3
. . . 2 - 3

6.
. . .
7. 10.1 . . .
10.1 . . .
8. . .
9. -
1.
1
2. 10.1 . . .
3. 10.1 . . .
4. . .
10.1 . . .
K-P-A

-
K
-
10.1
. . .
- -
5
P
-
-
-
- - 10.1
. . .
-
2
A
-
-
-
-
-
2

10 1
2, 4, 6, 8, 10, 12, 14, 16, 18,…
2
3, 6, 9, 12, 15, 18,… 3
6, 12, 18,... 2 3
6,12,18,... 2
3 2 3
. . . 2 3 . . . 2 3 6
10.1
. . .

. . .
. . .
1
. . . 8 12
8, 16, 24, 32, 40, 48,…
8
12, 24, 36, 48,… 12
24, 48, ... 8 12
24 8 12
. . . 8 12 24
24
2

. . . 6, 20 30
6, 20 30
6 = 2 3
20 = 2 2 5
30 = 2 3 5
6, 20 30
6, 20 30 2 2 3 5
= 60
. . . 6, 20 30 60
60

10.1
. . .
10 1
1.
1 . . . 4 6
4, 8, 12, 16, 20, 24, 28 32 36 …
4
6, 12, 18, 24, 30 36 42 …
6
12, 24, 36 ... 4
6
12 4 6
. . . 4 6 12
12

2 . . . 9 18
9, 18, 27, 36, 45 … 9
18, 36 54 … 18
18, 36 ... 9 18
18 9 18
. . . 9 18 18
18
2.
= 2 3 7
105 = 3 5 7
=
=
=
=
=

10.1
. . .
10 1
1.
1 . . . 4 6 2
…………………………………………………………………………
…………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………

2 . . . 9 18 2
…………………………………………………………………………
…………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
2.
2
…………………………………………………………………………
…………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………

………………………………………………………………………
……………………
………………………………………………………………………
……………………
2
…………………………………………………………………………
…………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
2
…………………………………………………………………………
…………………

………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………
………………………………………………………………………
……………………
______________________________________________ ______
___ ___________

:
4.
1 15 21 42 75
3 ) 1521 42 75
5 ) 5 7 14 25
7 ) 1 7 14 5
1 1 2 5
. . . 15 21 42 75 3 5 7 2 5
= 1 050
1 050
2 9 12 16
3 ) 912 16
2 ) 3 4 16
2 ) 3 2 8
3 1 4
. . . 9 12 16 3 2 2 3 4 =
144
144

1. . . . . . .
2. . . .
2.1 . . . 18, 30 36
2.2 . . .
. . .
1 18 30 36
2 ) 183036
3 ) 915 18
3 ) 3 5 6
1 5 2
. . . 18, 30 36 2 3 3 1 5 2
=180
180
2.3 . . . 2
1 . . . 24 40 ( 120)
2 . . . 28, 35 42 ( 420)

3. . . .
. . .
4. 3-4 11.1
. . . 11.1
. . .
5. . .
6. -
1.
1
2. 11.1 . . .
3. 11.1 . . .
11.1 . . .
K-P-A
-
K
-
- -
5

11.1
. . .
P
-
-
-
- -
11.1
. . .
-
2
A
-
-
-
-
-
2

11 1
1 15 21 42 75
3 ) 1521 42 75
5 ) 5 7 14 25
7 ) 1 7 14 5
1 1 2 5
. . . 15 21 42 75 3 5 7 2 5
= 1 050
1 050
2 9 12 16
3 ) 912 16
2 ) 3 4 16
2 ) 3 2 8
3 1 4
. . . 9 12 16 3 2 2 3 4 =
144
144
11.1 . . .

11.1 . . .
11 1
1.
1 12 18 20 24
2 ) 1218 20 24
2 ) 6 9 10 12
3 ) 3 9 5 6
1 3 5 2
12 18 20 24 2 2 3 3
5 2 = 360
360

11.1 . . .
2 15 25 60 75
3 ) 15 25 60 75
5 ) 5 25 20 25
5) 1 5 4 5
1 1 4 1
15 25 60 75 3 5 5 =
600
600
3 24 36 48 60 84
2 ) 2436 48 60 84
2 ) 1218 24 30 42
2) 6 9 12 15 21
3) 3 9 6 15 21
1 3 2 5 7
24 36 48 60 84
2 2 2 3 3 2 5 7 = 5 040
5 040
11 1

1.
1 12 18 20 24 2
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
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………………………………………………………………………
……………………
2 15 25 60 75 4
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………

………………………………………………………………………
……………………
………………………………………………………………………
……………………
3 24 36 48 60 84 4
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
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………………………………………………………………………
…………
………………………………………………………………………
…………
………………………………………………………………………
…………
………………………………………………………………………
…………
______________________________________________ ______
___ _________

12
1
1
1
14
1
3
K-P-A
:
. . .
:
1.
:

4.
1
15
4
12
5
8
5
. . . 8, 12 15 120
120
15
4
12
5
8
5
=
815
84
1012
105
158
155
=
30
4
30
24
30
5
=
120
32
120
50
120
75
=
120
325075
=
120
3225
=
120
57
=
40
19
40
19
2 6
8 10

6
8
10
6 8 10 120
120
120
1. . . .
2. . . .
2 - 3
1
15
2
5
4
-
6
1
. . . 5, 15 6 30
30
15
2
5
4
6
1
=
215
22
65
64
56
51
=
30
4
30
24
30
5
=
30
424
30
5

=
30
5
30
28
=
30
23
30
23
2 10, 20 30
. . . 10, 20 30
2 ) 10 2030
5 ) 5 1015
1 2 3
. . . 10, 20 30 2 5 2 3 = 60
60 1
1
3. 3-4
2
* 108, 54, 45 18
8
108, 54, 45 18
108, 54, 45 18 . . .
. . . 108, 54, 45 18

3 ) 108544518
3 ) 361815 6
3 ) 12 6 5 2
2 ) 4 2 5 2
2 1 5 1
. . . 108, 54, 45 18 3 3 3 2 2 5 =
540
108, 54, 45 18 540
108, 54, 45 18
8
540 + 8 = 548
548
4.
. .
5. 12.1
12.1
6.
7. -

1.
1
2. 12.1
3. 12.1
12.1
K-P-A
-
K
-
12.1
- -
5
P
-
-
-
- -
12.1
-
2

12 1
1
15
4
12
5
8
5
. . . 8, 12 15 120
12.1

120
15
4
12
5
8
5
=
815
84
1012
105
158
155
=
30
4
30
24
30
5
=
120
32
120
50
120
75
=
120
325075
=
120
3225
=
120
57
=
40
19
40
19
2 6
8
10
6
8
10
6 8 10 120
120
120
12.1

12 1
1. .
1
30
7
5
2
3
4
. . . 3, 5 30 30
30
30
7
5
2
3
4
=
30
7
65
62
103
104
=
30
4
30
24
30
5
=
30
7
30
12
30
40
=
30
71240
=
30
35
=
6
7
6
7

2
8
5
10
9
15
7
. . . 8, 10 15 120
120
8
5
10
9
15
7
=
158
155
1210
129
815
87
=
30
4
30
24
30
5
=
120
75
120
108
120
56
=
120
7510856
=
120
3356
=
120
89
120
89
3
9 12 20
9 12 20

2 ) 9 1220
2) 9 610
3) 9 3 5
3 1 5
. . . 9, 12 20 2 2 3 3 5 = 180
180
180
12.1
12 1
1. .
1
30
7
5
2
3
4
3
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………

………………………………………………………………………
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………………………………………………………………………
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………………………………………………………………………
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……………………
………………………………………………………………………
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………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
2
8
5
10
9
15
7
3
………………………………………………………………………
……………………
………………………………………………………………………
……………………

………………………………………………………………………
……………………
………………………………………………………………………
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………………………………………………………………………
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………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
3
9 12 20

4
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
……………………
………………………………………………………………………
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………………………………………………………………………
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………………………………………………………………………
……………………

………………………………………………………………………
……………………
………………………………………………………………………
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………………………………………………………………………
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………………………………………………………………………
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………………………………………………………………………
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………………………………………………………………………
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……………………
………………………………………………………………………
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………………………………………………………………………
……………………

______________________________________________ _
________ _________
13
1
1
1
14
1
“ ”

K-P-A
:
1.
:
1.
:
4.
x y
a x y
b x y
1.
“
”
a b =
x y

2. 2 1 30
42 2 30 42
3. 2 30
42 1 2
4.
1 12
40
12 = 2 2 3
40 = 2 2 2 5
12 40 2 2 =4
12 40 2 2 2 3 5 =120
4 120 = 480
12 40 = 480
2 . . . 24 36
2 ) 24 36
2 ) 12 18
3 ) 6 9
2 3
. . . 32 64 2 2 3 =12
32 64 2 2 2 3 3 = 72

12 72 = 864
24 36 = 864
3 5
195 15
m
5
195
15 m = 5 195
m =
15
1955
m = 65
65
65
5.
x y
a x
y
b x y
a b = x y

6. 13.1
13.1
7.
8. -
1.
1
2. 13.1
3. 13.1
13.1
K-P-A
-
K
-
- -

13.1 5
P
-
-
-
- -
13.1
-
2
A
-
-
-
-
-
2

13.1
13 1
1 18
48
18 = 2 3 3
48 = 2 2 2 2 3
18 48 2 3 =6
18 48 2 2 2 2 3
3 =144
6 144 = 864
18 48 = 864
2 . . . 27 45
3 ) 27 45
3 ) 9 15
3 5
. . . 27 45 3 3 = 9

13.1
27 45 3 3 3 5 = 135
9 135 = 1 215
27 45 = 1 215
3
4 168 24
m
4
168
24 m = 4 168
m =
24
1684
m = 28
28
28
13 1
1.
1 25
105
5 525 2 625 2 625
2 36 18 108 1 944 1 944

54
3 20
48
4 240 960 960
4 40
80
40 80 3 200 3 200
5 50
75
25 150 3 750 3 750
1.1
x y
a x y
b x y
a b = x y
2. 12 72
24
a
12
72
24 a = 12 72
a =
24
7212
a = 36

36
36
13.1
13 1
1. 5
1 25
105
2 36
54
3 20
48
4 40
80
5 50
75
1.1 ......................................
..........................................................................................................
..........................................................................................................
..........................................................................................................

..........................................................................................................
..........................................................................................................
..........................................................................................................
..........................................................................................................
..........................................................................................................
..........................................................................................................
....................................................................................................
2. 12 72
24
5
..........................................................................................................
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..........................................................................................................
..........................................................................................................

..........................................................................................................
................................................
______________________________________________ ______
___ _________
14
1
1
1
14
1
K-P-A
:
1. 50
2.
:
1.

:
4.
1.
2.
30
3.
4.
1.
1
2.
-
K-P-A

K
-
- -
50
P
-
-
-
- - -
2
A
-
-
-
-
-
2

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