Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

RULE. Divide the given number by any prime factor; divile the quotient in the same manner, and so continue the division until the quotient is a prime number. The several divisors and the last quotient will be the prime factors required.

Proof. The product of all the prime factors will be the given number.

EXAMPLES FOR PRACTICE. 1. What are the prime factors of 2150 ? 2. What are the prime factors of 2145 ? 3. What are the prime factors of 6300 ? 4. What are the prime factors of 21504? 5. What are the prime factors of 2366 ? 6. What are the prime factors of 1000 ? 7. What are the prime factors of 390625 ? 8. What are the prime factors of 999999 ? 143. If the prime factors of a number are small, as 2, 3, 5, 7, or 11, they may be easily found by the tests of divisibility, (136), or by trial. But numbers may be proposed requiring many trials to find their prime factors. This difficulty is obviated, within a certain limit, by the Factor Table given on pages 72, 73.

By prefixing each number in bold-face type in the column of Numbers, to the several numbers following it in the same division of the column, we shall form all the composite numbers less than 10,000, and not divisible by 2, 3, 5, 7, or 11; the numbers in the columns of Factors are the least prime factors of the numbers thus formed respectively. Thus, in one of the columns of Numbers we find 39, in bold-face type, and below 39, in the same column, is 77, which annexed to 39, forms 3977, a composite number. The least prime factor of this number is 41, which we find at the right of 77, in the column of Factors.

144. Hence, for the use of this table, we have the following

RULE. I. Cancel from the given number all factors less than 13, and then find the remaining factors by the table.

II. If any number less than 10,000 is not found in the table, and is not divisible by 2, 3, 5, 7, or 11, it is prime.

[ocr errors]
[blocks in formation]

99 29

43 29

83 17

09 31

69 13

83 71

41 23 59 17 69 53 87 13

Ol 17 23 13

83 13 89 19 91 47

57 31 69 13

63 13

83 19

01 19

43 43

13 13

21 23

49 13

01 19 89 17

37 31
61 31
13 17

31 47

07 31
+1 13

03 13
47 23
33 17
13 23

59 37
37 43

61 17 41 23

81 59
03 17
61 19

13 17
71 19
61 13

73 29

97 13
67 17
73 31

03 31

79 23
17 29
81 29

91 13
73 29

33 23 03 13

87 13

97 19

93 23

+9 29 51 59 63 19 67 23

19 17

[blocks in formation]
[blocks in formation]

33 37

27 53 31 31

33 13

39 19

69 17

1
11 17
79 37 | 41 17

17 53 77 31 9 17 13 57 19

51 13 97 43

13 19 27 29 2

61 37
77 13 34

21 29 47 47 | 91 29 21 13

31 61

67 67 52 47 13 43 23 15 20 25 87 29 03 41 93 17

69 1907 41 21 43 01 41 93 41 19 13 39

51 19 71 13 99 13 33 19 07 23 30 27 23 01 47 69 17 77 17

19 17 3 89 23 17 37

79 29 48 23 17 10 37 29

39 19 53 59 81 13 11 17 39 13 59 29 29 13 73 23

87 41 19 61 77 13 07 19 77 19

41 47 91 17 27 13 91 37 77 31

53 43

99 53 | 13 29 4 37 17 16 21

71 37 35 77 41 44 47 37
77 17

27 19 49 13 87 17 37 19 79 13 13 31 19 13

23 13
29 43 53 23

93 67 81 13 81 23 47 19 93 23 31 51 53 40

59
43

53 93 17 11 51 13 59 17 26

03 29

09 19 53 61 67 31 11 47 5 21 19 79 23

07 13 87 17 31 29 69 41 83 19 17 13 27 17

89 37

71 17 91 67 21 17 29 23 17 31 91 19 83 37 27 37

99 59 43 13 89 67 97 59 29 73 57 13 17 97 13 41 19 33 13 36 61 31 45 49 51 19 59 19 03 13 22

01 13

11 1301 13 53 53 89 29 01 31 27

13 17 59 23 89 19 12 17 17 09 47 51 23 29 19 87 61

27 13 6

97 17 41 19 79 13 53 13 41

53 29 81 17 77 19 29 17 63 41

67 19

59 47 97 19 | 89 17 67 23 69 29 57 37 97 23 79 13

73 17 50 54 61 13 63 31 73 47 32

41 41 77 23

17 29 29 61 97 17 71 31 18 79 43 28 11 13 37

29 47 47 13 7 73 19 07 13

59 53 03 19 13 17 23 23

21 61

46 13 13 19 17

31 19

37 37 83 47 33 31 29 31 27 13

87 53

63 61 91 17 67 13

67 47 77 29

89 59

69 37 43 17 49 43

55 53 13 69 19 93 13 19 19 53 17

73 13 87

42 61 59

51
57 23
69 23 81 43

81 19

67 13 11 19 39 29 8 19 24 99 13 33

91 17 37 19 17 19 07 29 29

87 43

19 31 41 29 19 19 37 47 38

93 13 41 53 61 67 51 23 91 13 21 17 1941 21 23 41 13 | 09 13 43 99 37 71 13 14 49 31 23 37

11 37 03 13
47

87 37 07 59

61 13 97 29

[blocks in formation]
[blocks in formation]

11 23

[blocks in formation]
[blocks in formation]
[blocks in formation]
[blocks in formation]

37 79 85

56 60
51 13 61 53

53 19
67 13 61 47

63 79
23 19 67 29
79 29

79 13 69 13
89 83
89 37 77

81 83 71 73 97 93 61 91 23 09 13

91 17
59 73
97 73 69
73

83 59
79 61 89

99 17 33 43 71 13 99 67 01 67 03 67 39 71 81

03 29 93

07 17 77 59 65

47 61

89 13

09 59 01 71 27 71 81 1361 29 13 19 13

17 37
03 17
31 29 27 17

27 79 13 67 61 43 57 27 61 43 53

47 23 07 13

61 17 81 31

09 67 57 13

47 13 19 29

31 19 59 17

97 97 23 59 37 17 41 31 89 29 67 53

49 83 77 47

99 41 57 79 70 73 73 78

77 13 51 17

79 83 98 61 61 83 29 03 47 79 47 01 29 89 19

89 89 89 41 67 73 69 31

09 43
87 83
07 37 82

93 17 94
79 37 66
31 79
11 73

79 23 90 07 23
17 71

47 43 77 53 91 41 17 13 37 31

74
31 41 07 29

93 13
19 29

53 59 58 62

37 17
86 47 83

69 71
27 13
67 37 21 41

27 19 11 79

61 13

81 19 81 41 81 73 23 13 59 29 49 73 21 37 71 47 87 53 37 13 47 17 87 19 29 17

33 89 73 43 95

99 19 49 61

57 23 39 53

03 13 99 67 59

53 29 97 53 79 17 51 41 83 31 09 37
83 61
99 31 63 17 179

89 61 17 31
59 89 19 97 37 71 71 31 13 41 83
09 19 63 67 11 13 93 59 21 89 03 19

43 61 13 59 07 19 23 17 75

21 53 87

53 41 17 61 19 71 31 53 41 37

43 13 33 13 11 31 21 31 | 31 13

57 73 33 1741 17 49 17 57 17

61 19 63 13

47 17

67 89 77 61 83 67 47 19 | 83 13

71 67 69 13
57 61

69 53 89 43
59 59 64
67 67 | 71 71

59 13

77 67 79 67 93 53 63 67 01 37 73 13 81 43 76 81 23 81 17 91 59 69 4703 19

91 61 83 83

97 17 96 77 43

92 07 43 68 72 83 31 17 17

17 59 89 53 31 59

59 13 13 47 51 53 57 13 23 71

59 47

71 19 27 23

17 19 57 17 39 47

63 59

73 17 59 19

73 19

41 23
43 17
33 29

83 23 01 17 03 13

77 13
77 19

53 79

47 13
19 13
63 23

97 43
09 71
87 71

51 83
71 43

87 37

01 89 11 31

77 41

73 37 87 13

03 31 17 41 31 37 93 43

29 59 49 23 27 17

83 17 29 13

19 23

07 41

31 37 13 31 13 71

31 47

97 29

51 23 71 53 09 23 69 17

22 19

63 13

07 47
11 17

71 19
43 17

73 29 99 41

39 41
49 29

53 47
07 31
33 47
53 17

53 31
09 41

67 17
83 43

63 37
39 13
73 19

59 41
87 13

83 13 13 29

09 17 57 43

27 31 29 17 57 47

67 13

41 13 59 13 93 19

01 59

09 97 91 19

03 13

87 31 13 13

51 13 71 29

33 13

97 13
13 17
87 23

69 17 73 23

13 43
61 23
09 31

93 13 23 37

49 47 31 19 09 37

41 29 33 23

71 17

51 37 33 19

77 29

13 23 39 17

93 41
39 43
91 13

17 47 91 43 41 79 97 47

99 43
53 17

23 89 37 19 93 71 53 13 83 41

01 19

29 13 99 17

83 19

53 37

13 13 39 17

59 23 31 23

57 19 63 73

71 13

39 13 11 23

01 13
17 23

79 17 39 31

43 41

71 17 53 23

19 73 39 23

41 19

49 13
31 17
59 19

91 97
43 19
67 31
73 31

97 13 71 23

51 43 41 13

57 29
69 67

93 29

99 29
97 71
79 79
97 19

07 13
99 13
99 23
13 23
99 19

99 37
19 19

71 13 91

88

11 61 01 19 27 29 80 84

21 19 23 31 31 13 03 53 01 31 93 13 37 41 41 13 33 17 21 13 11 41

41 31

01 13
17 13

37 23

09 23 09 13

23 23 43 37

47 41

OPERATION.

37 x 53,

1. Resolve 1961 into its prime factors.

Analysis. Cutting off the two 1961 ; 37 = 53

right hand figures of the given 1961

Ans. number, and referring to the table,

column No., we find the other part, 19, in bold-face type; and under it, in the same division of the column, we find 61, the figureş cut off; at the right of 61, in column Fac., we find 37, the least prime factor of the given number. Dividing by 37, we obtain 53, the other factor. 2. Resolve 188139 into its prime factors.

ANALYSIS We find by trial 3188139

that the given number is divisible

by 3 and 7; dividing by these fac7 62713

tors, we have for a quotient 8959. 8959

By referring to the factor table, 17 527

we find the least prime factor of

this number to be 17; dividing by 31

17, we have 527 for a quotient, 3x7x17x17x31, Ans. Referring again to the table, we

find 17 to be the least factor of 527, and the other factor, 31, is prime.

OPERATION.

17

EXAMPLES FOR PRACTICE.

Ans. 2, 13, 727.

1. Resolve 18902 into its prime factors. 2. Resolve 352002 into its prime factors. 3. Resolve 6851 into its prime factors. 4. Resolve 9367 into its prime factors. 5. Resolve 203566 into its prime factors. 6. Resolve 59843 into its prime factors. 7. Resolve 9991 into its prime factors. 8. Resolve 123015 into its prime factors. 9. Resolve 893235 into its prime factors. 10. Resolve 390976 into its prime factors. 11. Resolve 225071 into its prime factors. 12. Resolve 81770 into its prime factors. 13. Resolve 6409 into its prime factors. 14. Resolve 178296 into its prime factors. 15. Resolve 714210 into its prime factors.

CASE II.

145. To find all the exact divisors of a number.

It is evident that all the prime factors of a number, together with all the possible combinations of those prime factors, will constitute all the exact divisors of that number, (142, II).

1. What are all the exact divisors of 360 ?

[ocr errors]
[ocr errors]

360

1 3

9 Ans.

5 15

OPERATION.
1 X 2 X 2 X 2 X 3 X 3 X 5.

2 4 8 Combinations of 1 and 2.
6 12
24

16 1 and 2 and 3.
18 36 72
10 20 40

66 1 and 2 and 5.
30 60 120

16 1 and 2 and 3 and 5 90 180

360)

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

45 ,

ANALYSIS.

By Case I we find the prime factors of 360 to be 1, 2. 2, 2, 3, 3, and 5. As 2 occurs three times as a factor, the different combinations of 1 and 2 by which 360 is divisible will be 1, 1x2=2, 1x 2 X 2=4, and 1 X 2 X 2 X 2=8; these we write in the first line. Multiplying the first line by 3 and writing the products in the second line, and the second line by 3, writing the products in the third line, we have in the first, second and third lines all the different combinar tions of 1, 2, and 3, by which 360 is divisible. Multiplying the first, second and third lines by 5, and writing the products in the fourth, fifth and sixth lines, respectively, we have in the six lines together every combination of the prime factors by which the given number, 360, is divisible.

Hence the following
RULE. I. Resolve the given number into its prime factors.

II Form a series having 1 for the first term, that prime factor which occurs the greatest number of times in the given number for the second term, the square of this factor for the third term, and so on, till a term is reached containing this factor as many times as it occurs in the given number.

III. Multiply the numbers in this line by another factor, ind these results by the same factor, and so on, as many times as this factor occurs in the given number.

« ΠροηγούμενηΣυνέχεια »