Euclidian Algorithm Loves One

8/11/2019 Euclidian Algorithm Loves One

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II Statement of the problem

A. Situation

Use of consecutive odd numbers, consecutive even numbers and Euclidian

Algorithm.

B. Problem

1.

Is there any pattern or common value of GCD arrived when you use twoconsecutive even numbers and two consecutive odd numbers in Euclidian

Algorithm.

2. How does the multiplier of each second step differ or relate itself when it

classified as even or odd number?

3. Is there any pattern or common value of GCD arrived when you use two

consecutive even numbers and two reversed consecutive odd numbers in

Euclidian Algorithm.

4. How does the multiplier of each second step differ or relate itself when it

classified as even or odd number?

III Data gathered

DATA-1

GCDs of the first 55 pairs of numbers (two consecutive even numbers and two

consecutive odd numbers) used in Euclidian algorithm.

GCD (24,13)

24=1.13+11

13=1.11+2

11=5.2+12=2.1+0

GCD=1

GCD (46,35)

46=1.35+11

35=3.11+2

11=5.2+1

2=2.1+0

GCD=1

GCD (68,57)68=1.57+11

57=5.11+2

11=5.2+1

2=2.1+0

GCD=1

GCD (810,79)

810=10.79+20

79=3.20+19

20=1.19+1

19=19.1+0GCD=1

GCD (1012,911)

1012=1.911+101

911=9.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (1214,1113)

1214=1.1113+101

1113=11.103+101

103=1.101+2101=50.2+1

2=2.1+0

GCD=1

GCD (1416,1315)

1416=1.1315+101

1315=13.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (1618, 1517)

1618=1.1517+101

1517=15.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (1820,1719)

1821=1.1719+101

1719=17.101+2

101=50.2+12=2.1+0

GCD=1

GCD (2022,1921)

2022=1.1921+101

1921=19.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (2224,2123)

2224=1.2123+101

2123=21.101+2

101=50.2+12=2.1+0

GCD=1

GCD (2426,2325)

2426=1.2325+101

2325=23.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (2628,2527)2628=1.2527+101

2527=25.101+2

101=50.2

2=2.1+0

GCD=1

GCD (2830,2729)

2830=1.2729+101

2729=27.101+2

101=50.2+1

2=2.1+0GCD=1

GCD (3032,2931)

3032=1.2933+101

2933=29.101+2

101=50.2+1

2=2.1+0

GCD=1


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GCD (8082,7981)

8082=1.7981+101

7981=79.101+2

101=50.2+1

2=2.1+0GCD=1

GCD (8284,8183)

6264=1.8183+101

8183=81.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (8486,8385)

8486=1.8385+101

8385=83.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (8688,8587)

8688=1.8587+101

8587=85.101+2101=50.2+1

2=2.1+0

GCD=1

GCD (8890,8789)

8890=1.8789+101

8789=87.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (9092,8991)

9091=1.8991+101

8991=89.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (9294,9193)

9294=1.9193+101

9193=91.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (9496,9395)

9494=1.9395+101

9395=93.101+2

101=50.2+1

2=2.1+0GCD=1

GCD (9698,9597)

9698=1.9597+101

9597=95.101+2

101=50.2+1

2=2.1+0

GCD=1

GCD (98100,9799)

98100=10.9799+110

9799=89.110+9

110=12.9+2

9=4.2+1

2=2.1+0

GCD=1

GCD (100102,99101)

100102=1.99101+100199101=99.1001+2

101=500.2+1

2=2.1+0

GCD=1

GCD (102104,101103)

102104=1.101103+1001

101103=101.1001+2

1001=500.2+1

2=2.1+0GCD=1

GCD (104106,103105)

104106=1.103105+1001

103105=103.1001+2

1001=500.2+1

2=2.1+0

GCD=1

GCD (106108,105107)

106108=1.105107+1001

105107=105.1001+2

1001=500.2+1

2=2.1+0

GCD=1

GCD (108110,107109)

108110=1.107109+1001

107109=107.1001+2

1002=500.2+1

2=2.1+0GCD=1

GCD (110112,109111)

110112=1.109111+1001

109111=109.1001+2

1001=500.2+1

2=2.1+0

GCD=1


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DATA-2Value of the multiplier of the first 55 pairs of numbers (two consecutive even numbers and

two consecutive odd numbers) used in Euclidian algorithm.

GCD(Ee,Oo) Multiplier(M) Even/odd

GCD (24,13) 1 Odd

GCD (46,35) 3 OddGCD (68,57) 5 Odd

GCD (810,79) 3 Odd

GCD (1012,911) 9 Odd

GCD (1214,1113) 11 Odd

GCD (1416,1315) 13 Odd

GCD (1618,1517) 15 Odd

GCD (1820,1719) 17 Odd

GCD (2022,1921) 19 Odd

GCD (2224,2123) 21 Odd

GCD (2426,2325) 23 Odd

GCD (2628,2527) 25 OddGCD (2830,2729) 27 Odd

GCD (3032,2931) 29 Odd

GCD (3234,3133) 31 Odd

GCD (3436,3335) 33 Odd

GCD (3638,3537) 35 Odd

GCD (3840,3739) 37 Odd

GCD (4042,3041) 39 Odd

GCD (4244,4143) 41 Odd

GCD (4446,4345) 43 Odd

GCD (4648,4547) 45 OddGCD (4850,4749) 47 Odd

GCD (5052,4951) 49 Odd

GCD (5254,5153) 51 Odd

GCD (5456,5355) 53 Odd

GCD (5658,5557) 55 Odd

GCD (5860,5759) 57 Odd

GCD 6062,5961) 59 Odd

GCD (6264,6163) 61 Odd

GCD (6466,6365) 63 Odd

GCD (6668,6567) 65 Odd

GCD (6870,6769) 67 Odd

GCD (7072,6971) 69 Odd

GCD (7274,7173) 71 Odd

GCD (7476,7375) 73 Odd

GCD (7678,7577) 75 Odd

GCD (7880,7779) 77 Odd

GCD (8082,7981) 79 Odd

GCD (8284,8183) 81 Odd

GCD (8486,8385) 83 Odd

GCD (8688,8587) 85 Odd

GCD (8890,8789) 87 OddGCD (9092,8991) 89 Odd

GCD (9294,9193) 91 Odd

GCD (9496,9395) 93 Odd

GCD (9698,9597) 95 Odd

GCD (98100,9799) 89 Odd

GCD (100102,99101) 99 Odd

GCD (102104,101103) 101 Odd

GCD (104106,103105) 103 Odd

GCD (106108,105107) 105 Odd

GCD (108110,107109) 107 Odd

GCD (110112,109111) 109 Odd


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DATA-3

GCDs of the first 55 pairs of numbers (two consecutive even numbers and two

reversed consecutive odd numbers) used in Euclidian algorithm.

GCD (24,31)

31=1.24+724=3.7+3

7=2.3+1

3=3.1+0

GCD=1

GCD (46,53)

53=1.46+7

46=6.7+4

7=1.4+3

4=1.3+13=3.1+0

GCD=1

GCD (68,75)

75=1.68+7

68=9.7+5

7=1.5+2

5=2.2+1

2=2.1+0

GCD=1

GCD (810,97)

810=8.97+34

97=2.34+29

34=1.29+5

29=5.5+4

5=1.4+1

4=4.1+0

GCD=1

GCD (1012,119)

1012=8.119+60

119=1.60+59

60=1.59+1

59=1.59+0

GCD=1

GCD (1214,1311)

1311=1.1214+97

1214=12.97+50

97=1.50+47

50=1.47+3

47=15.3+2

3=1.2+1

2=2.1+0

GCD=1

GCD (1416,1513)

1513=1.1416+971416=14.97+58

97=1.58+39

58=1.39+19

39=2.19+1

19=119.1+0

GCD=1

GCD (1618,1715)

1715=1.1618+100

1618=16.100+18

100=5.18+1018=1.10+8

10=1.8+2

8=4.2+0

GCD=1

GCD (1820,1917)

1917=1.1820+97

1820=18.97+74

97=1.74+23

74=3.23+523=4.5+3

5=1.3+2

3=1.2+1

2=2.1+0

GCD=1

GCD (2022,2119)

2119=1.2224+97

2224=20.97+82

97=1.82+1582=5.15+7

15=2.7+1

7=7.1+0

GCD=1

GCD (2224,2321)

2321=1.2224+97

2224=122.97+90

97=1.90+7

90=12.7+6

7=1.6+1

6=6.1+0

GCD=1

GCD (2426,2523)

2523=1.2426+97

2426=25.97+1

97=97.1+0

GCD=1

GCD (2628,2725)2725=1.2628+97

2628=27.97+9

97=10.9+7

9=1.7+

7=3.2+1

2=2.1+0

GCD=1

GCD (2830,2927)

2927=1.2830+972830=29.97+17

97=5.17+12

17=1.12+5

12=2.5+2

5=2.2+1

2=2.1+0

GCD=1

GCD (3032,3129)

3129=1.3032+973032=31.97+25

97=3.255+22

25=1.22+3

22=7.3+2

3=1.2+1

2=2.1+0

GCD=1

GCD (3234,3331)

3331=1.3234+973234=33.97+33

97=2.33+31

33=1.31+2

31=15.2+1

2=2.1+0

GCD=1

GCD (3436,3533)

35333=1.3436+97

3436=35.97+41

97=2.41+15

41=2.15+11

15=1.11+4

11=2.4+3

4=1.3+1

3=3.1+0

GCD=1


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GCD (3638,3735)

3735=1.3638+97

3638=37.97+49

97=1.49+48

49=1.48+1

48=48.1+0GCD=1

GCD (3840,3937)

3037=1.3840+97

3840=39.97+57

097=1.57+40

57=1.40+17

40=2.17+6

17=2.6+5

6=1.5+15=5.1+0

GCD=1

GCD (4042,4139)

4139=1.4042+97

4042=41.97+65

97=1.65+32

65=2.32+1

32=32.1+0

GCD=1

GCD (4244,4341)

4341=1.4244+97

4244=43.97+73

97=1.73+24

73=3.24+1

24=24.1+0

GCD=1

GCD (4446,4543)

4543=1.4446+97

4446=45.97+81

97=1.81+16

81=5.16+1

16=16.1+0

GCD=1

GCD (4648,4745)

4745=1.4648+97

4648=47.97+89

97=1.89+8

89=11.8+1

88=88.1+0

GCD=1

GCD (4850,4947)

4947=1.4850+97

4850=50.97+0

GCD=1

GCD (5052,5149)

5149=1.5052+97

5152=52.97+8

97=12.8+1

8=8.1+0

GCD=1

GCD (5254,5351)

5351=1.5254+97

5254=54.97+16

97=6.16+1

16=16.1+0

GCD=1

GCD (5456,5553)5553=1.5456+97

5456=56.97+24

97=4.24+1

24=24.1+0

GCD=1

GCD (5658,5755)

5755=5658.97

5658=58.97+32

97=3.32+132=32.1+0

GCD=1

GCD (5860,5957)

5957=1.5860+97

5860=60.97+40

97=2.40+17

40=2.17+6

17=2.6+5

6=1.5+15=5.1+0

GCD=1

GCD (6062,6159)

6159=1.6162+97

6062=62.97+48

97=2.48+1

48=48.1+0

GCD=1

GCD (6264,6361)

6361=1.6264+97

6264=64.97+56

97=1.56+41

56=1.41+15

41=2.15+11

15=1.11+4

11=2.4+3

4=1.3+1

3=3.1+0

GCD=1

GCD (6466,6563)

6563=1.6466+97

6466=66.97+64

97=1.64+33

64=1.33+31

33=1.31+231=15.2+1

2=2.1+0

GCD=1

GCD (6668,6765)

6765=1.6668+97

6668=68.97+72

97=1.72+25

72=2.25+22

25=1.22+322=7.3+1

3=3.1+0

GCD=1

GCD (6870,6967)

6967=1.6870+97

6870=70.97+80

97=1.80+17

80=4.17+1217=1.12+5

12=2.5+2

5=2.2+1

2=2.1+0

GCD=1

GCD (7072,7169)

7169=1.7072+97

7072=72.97+88

97=1.88+988=9.9+7

9=1.7+2

2=3.2+1

2=2.1+0

GCD=1

GCD (7274,7371)

7371=1.7274+97

7274=74.97+96

97=1.96+1

96=96.1+0

GCD=1

GCD (7476,7573)

7573=1.7476+97

7476=77.97+7

97=13.7+6

7=1.6+1

6=6.1+0

GCD=1


8/18

GCD (7678,7775)

7775=1.7678+97

7678=79.97+15

97=6.15+7

15=2.7+1

7=7.1+0GCD=1

GCD (7880,7977)

7977=1.7880+97

7880=81.97+23

97=4.23+5

23=4.5+3

5=1.3+2

3=1.2+1

2=2.1+0

GCD=1

GCD (8082,8179)

8179=1.8082+97

8082=83.97+31

97=3.31+4

31=7.4+3

4=1.3+1

3=3.1+0

GCD=1

GCD (8284,8381)8381=1.8284+97

8284=85.97+39

97=2.39+19

39=2.19+1

19=19.1+0

GCD=1

GCD (8486,8583)

8583=1.8486+97

8486=87.97+47

97=2.47+347=15.3+2

3=1.2+1

2=2.1+0

GCD=1

GCD (8688,8785)

8785=1.8688+97

8688=89.97+55

97=1.55+42

55=1.42+13

42=3.13+3

13=4.3+1

3=3.1+0

GCD=1

GCD (8890,8987)

8987=1.8890+97

8890=91.97+63

97=1.63+34

63=1.34+29

34=1.29+5

29=5.5+4

5=1.4+1

4=4.1+0

GCD=1

GCD (9092,9189)

9189=1.9092+97

9092=93.97+7197=1.71+26

71=2.26+19

26=1.19+7

19=2.7+5

7=1.5+2

5=2.2+1

2=2.1+0

GCD=1

GCD (9294,9391)

9391=1.9294+979294=95.97+79

97=1.79+18

79=4.18+7

18=2.7+4

7=1.4+3

4=1.3+1

3=3.1+0

GCD=1

GCD (9496,9593)

9593=1.9496+979496=97.97+87

97=1.87+10

87=8.10+7

10=1.7+3

7=2.3+1

3=3.1+0

GCD=1

GCD (9698,9795)

9795=1.9698+97

9698=99.97+9597=1.95+2

95=47.2+1

2=2.1+0

GCD=1

GCD (98100,9997)

98100=9.9997+8127

9997=1.8127+1870

8127=4.1870+647

8170=2.647+576

647=1.576+71

576=8.71+8

71=8.8+7

8=1.7+1

7=7.1+0

GCD=1

GCD (100102,10199)

100102=9.10199+8311

10199=1.8311+1888

8311=4.1888+759

1888=2.759+370

759=2.370+19

370=19.19+9

19=2.9+1

9=9.1+0

GCD=1

GCD (102104,103101)102101=1.102104+997

102104=102.997+410

997=2.410+177

410=2.177+56

177=3.56+9

56=6.9+2

9=4.2+1

2=2.1+0

GCD=1

GCD (104106,105103)

105103=1.104106+997

104106=104.997+418

997=2.418+161

418=2.161+96

161=1.96+65

96=1.65+31

65=2.31+3

31=10.3+1

3=3.1+0

GCD=1

GCD (106108,107105)

107105=1.106108+997

106108=106.997+426

997=2.426+145

426=2.245+136

145=1.136+9

136=15.9+1

9=9.1+0

GCD=1

GCD (108110,109107)

109107=1.108110+997

108110=108.997+434

997=2.434+129

434=3.129+47

129=2.47+35

47=1.35+12

35=2.12+11

12=1.11+1

11=11.1+0

GCD=1

GCD (110112,111109)

111109=1.110112+997

110112=110.997+442

997=2.442+113

442=3.113+103

113=1.103+10

103=10.10+3

10=3.3+1

3=3.1+


9/18

DATA 4

Values of the multiplier (second step) of the first 55 pairs of numbers (two

consecutive even numbers and two reversed consecutive odd numbers).

GCD(Ee.oO) Multiplier(M) Even/odd

GCD(24,31) 3 Odd

GCD(46,53) 6 even

GCD(68,57) 9 Odd

GCD(810,97) 2 even

GCD(1012,119) 1 Odd

GCD(1214,1311) 12 even

GCD(1416,1513) 14 evenGCD(1618,1715) 16 even

GCD(1820,1917) 18 even

GCD(2022,2119) 20 even

GCD(2224,2321) 22 even

GCD(2426,2523) 25 Odd

GCD(2628,2725) 27 Odd

GCD(2830,2927) 29 Odd

GCD(3032,3129) 31 Odd

GCD(3234,3331) 33 Odd

GCD(3436,3533) 35 Odd

GCD(3638,3735) 37 Odd

GCD(3840,3937) 39 Odd

GCD(4042,4139) 41 Odd

GCD(4244,4341) 43 Odd

GCD(4446,4543) 45 Odd

GCD(4648,4745) 47 Odd

GCD(4850,4947) 50 even

GCD(5052,5149) 52 even

GCD(5254,5351) 54 even

GCD(5456,5553) 56 even

GCD(5658,5755) 58 even

GCD(5860,5957) 60 even

GCD(6062,6159) 62 evenGCD(6264,6361) 64 even

GCD(6466,6563) 66 even

GCD(6668,6765) 68 even

GCD(6870,6967) 70 even

GCD(7072,7169) 72 even

GCD(7274,7371) 74 even

GCD(7476,7573) 77 Odd

GCD(7678,7775) 79 Odd

GCD(7880,7977) 81 Odd

GCD(8082,8179) 83 Odd

GCD(8284,8381) 85 Odd

GCD(8486,8583) 87 Odd

GCD(8688,8785) 89 Odd

GCD(8890,8987) 91 Odd

GCD(9092,9189) 93 Odd

GCD(9294,9391) 95 Odd

GCD(9496,9593) 97 Odd

GCD(9698,9795) 99 Odd

GCD(98100,9997) 1 Odd

GCD(100102,10199) 1 Odd

GCD(102104,103101) 102 even

GCD(104106,105103) 104 even

GCD(106108,107105) 106 evenGCD(108110,109107) 108 even

GCD(110112,111109) 110 even


10/18


11/18

C. Conjectures

The following conjectures are arrived after the study of the data gathered by the

investigator.

1. The GCDs of the two consecutive even numbers and two consecutive odd

numbers ended with only one value which is one (1).

GCD (Ee,Oo)

Where:

- E is the initial even number

-e is the preceded even number

-O is the initial odd number

-o is the preceded odd number

2. The multiplier of each second step of two consecutive enen numbers and two

consecutive odd numbers ended with all odd numbers.

Step 2: Oo=

Where:

-Oo is two consecutive odd numbers

- is the multiplier

-is the remainder of step 1


3. The GCDs of the two consecutive even numbers and two reversed consecutive

odd


GCD (Ee, oO)

Where:





4. The multiplier of the two consecutive even numbers and two

reversed consecutive odd numbers ended with a sequence of odd andeven numbers but not in a regular manner(which means they appear in a

sequence started with an odd and then with an even number but they appear in

large quantities.

Step 2: oO=

Where:

-oO is two reversed consecutive odd numbers

- is the multiplier




12/18

III. Verifying conjecturesOn conjecture 1

Attempts Ee Oo GCD

1 24 13 1

2 46 35 1

3 68 57 1

4 810 79 1

5 1012 911 1

6 1214 1113 1

7 1416 1315 1

8 1618 1517 1

9 1820 G1719 110 2022 1921 1

11 2224 2123 1

12 2426 2325 1

13 2628 2527 1

14 2830 2729 1

15 3032 2931 1

16 3234 3133 1

17 3436 3335 1

18 3638 3537 1

19 3840 3739 1

20 4042 3041 121 4244 4143 1

22 4446 4345 1

23 4648 4547 1

24 4850 4749 1

25 5052 4951 1

26 5254 5153 1

27 5456 5355 1

28 5658 5557 1

29 5860 5759 1

30 6062 5961 131 6264 6163 1

32 6466 6365 1

33 6668 6567 1

34 6870 6769 1

35 7072 697 1

36 7274 7173 1

37 7476 7375 1

38 7678 7577 1

39 7880 7779 1

40 8082 7981 1

41 8284 8183 142 8486 8385 1

43 8688 8587 1

44 8890 8789 1

45 9092 8991 1

46 9294 9189 1

47 9496 9395 1

48 9698 9597 1

49 98100 9799 1

50 100102 99101 1

51 102104 101103 152 104106 103105 1


13/18

53 106108 105107 1

54 108110 107109 1

55 110112 109111 1

On Conjecture 2

Attempts Oo= Multiplier(M)

1 13=1.11+2 1

2 35=3.11+2 3

3 57=5.11+2 5

4 79=3.20+19 3

5 911=9.101+2 9

6 1113=11.103+101 11

7 1315=13.101+2 13

8 1517=15.101+2 159 1719=17.101+2 17

10 1921=19.101+2 19

11 2123=21.101+2 21

12 2325=23.101+2 23

13 2527=25.101+2 25

14 2729=27.101+2 27

15 2933=29.101+2 29

16 3133=31.101+2 31

17 3335=33.101+2 33

18 3537=35.101+2 3519 3739=37.101+2 37

20 3941=39.101+2 39

21 4143=41.101+2 41

22 4345=43.101+2 43

23 4547=45.101+2 45

24 4749=47.101+2 47

25 4951=49.101+2 49

26 5153=51.101+2 51

27 5355=53.101+2 53

28 5557=55.101+2 55

29 5759=57.101+2 5730 5961=59.101+2 59

31 6163=61.101+2 61

32 6365=63.101+2 63

33 6567=65.101+2 65

34 6769=67.101+2 67

35 6971=69.101+2 69

36 7173=71.101+2 71

37 7375=73.101+2 73

38 7577=75.101+2 75

39 7779=77.101+2 7740 7981=79.101+2 79

41 8183=81.101+2 81

42 8385=83.101+2 83

43 8587=85.101+2 85

44 8789=87.101+2 87

45 8991=89.101+2 89

46 9193=91.101+2 91

47 9395=93.101+2 93

48 9597=95.101+2 95

49 9799=89.110+9 97

50 99101=99.1001+2 99


14/18

51 101103=101.1001+2 101

52 103105=103.1001+2 103

53 105107=105.1001+2 105

54 107109=107.1001+2 107

55 109111=109.1001+2 109

On conjecture 3

Attempts Ee oO GCD

1 24 31 1

2 46 5 3 1

3 68 75 1

4 810 97 1

5 1012 119 1

6 1214 1311 1

7 1416 15 13 1

8 1618 1715 1

9 1820 1917 1

10 2022 2119 1

11 2224 2321 1

12 2426 2523 1

13 2628 2725 1

14 2830 2927 1

15 3032 3129 1

16 3234 3331 117 3436 3533 1

18 3638 3735 1

19 3840 3937 1

20 4042 4139 1

21 4244 4341 1

22 4446 4543 1

23 4648 4745 1

24 4850 4947 1

25 5052 5149 1

26 5254 5351 1

27 5456 5553 1

28 5658 5755 1

29 5860 5957 1

30 6062 6159 1

31 6264 6361 1

32 6466 6563 1

33 6668 6765 1

34 6870 6967 1

35 7072 7169 1

36 7274 7371 1

37 7476 7573 138 7678 7775 1

39 7880 7977 1

40 8082 8179 1

41 8284 8381` 1

42 8486 8583 1

43 8688 8785 1

44 8890 8987 1

45 9092 9189 1

46 9294 8991 1

47 9496 9593 1

48 9698 9795 1


15/18

49 98100 9997 1

50 100102 10199 1

51 102104 103101 1

52 104106 105103 1

53 106108 107105 1

54 108110 109107 155 110112 111109 1

On conjecture 4

Attempts oO= Multiplier(M)

1 24=3.7+3 3

2 46=6.7+4 6

3 68=9.7+5 9

4 97=2.34+29 2

5 119=1.60+59 1

6 1214=12.97+50 12

7 1416=14.97+58 14

8 1618=16.100+18 16

9 1820=18.97+74 18

10 2024=20.97+82 20

11 2224=122.97+90 22

12 2426=25.97+1 25

13 2628=27.97+9 27

14 2830=29.97+17 29

15 3032=31.97+25 31

16 3234=33.97+33 3317 3436=35.97+41 35

18 3638=37.97+49 37

19 3638=37.97+49 39

20 3840=39.97+57 41

21 4042=41.97+65 43

22 4244=43.97+73 45

23 4446=45.97+81 47

24 4648=47.97+89 50

25 4850=50.97+0 52

26 5052=52.97+8 54

27 5254=54.97+16 56

28 5456=56.97+24 58

29 5658=58.97+32 60

30 5860=60.97+40 62

31 6062=62.97+48 64

32 6264=64.97+56 66

33 6466=66.97+64 68

34 6668=68.97+72 70

35 6870=70.97+80 72

36 7072=72.97+88 74

37 7274=74.97+96 7738 7476=77.97+7 79

39 7678=79.97+15 81

40 7880=81.97+23 83

41 8082=83.97+31 85

42 8284=85.97+39 87

43 8486=87.97+47 89

44 8688=89.97+55 91

45 8890=91.97+63 93

46 9092=93.97+71 95

47 9294=95.97+79 97

48 9496=97.97+87 99


16/18

49 9698=99.97+95 1

50 9997=1.8127+1870 1

51 10199=1.8311+1888 102

52 104106=104.997+418 104

53 106108=106.997+426 106

54 108110=108.997+434 10855 110112=110.997+442 110

IV. Justification

Using the Euclidian Algorithm is an efficient method for computing the

greatest common divisor (GCD), also known as the greatest common factor

(GCF) or highest common factor (HCF).

Let GCD (Ee,Oo)

Where:





Please refer to DATA 1

Let GCD (Ee, oO)

Where:

- E is the initial even number-e is the preceded even number



V Summary

This mathematical investigation on how an even and an odd numbers played by the

investigator through Euclidian Algorithm I order to arrive a certain pattern , the same GCD,

and an ideas of having odd numbers on multipliers even if how we pair the numbers.

More try outs done to serve as a proof.

1. DATA 1 is made to find the GCD when we pair the two consecutive even numbers to

the two consecutive odd numbers.

2. DATA 2 is made to determine on how does the multiplier of each second step differ

or relate itself when it classified as even or odd number.

3. DATA 1 is made to find the GCD when we pair the two consecutive even numbers to

the two reversed consecutive odd numbers.

4. DATA 2 is made to determine on how does the multiplier of each second step differ

or relate itself when it classified as even or odd number.

After data set were gathered, the conjecture arrived were

1. The GCDs of the two consecutive even numbers and two consecutive odd


GCD (Ee,Oo)

Where:




http://en.wikipedia.org/wiki/Greatest_common_divisorhttp://en.wikipedia.org/wiki/Greatest_common_divisor


17/18


18/18

Philippine Normal University

NATIONAL CENTER FOR TEACHER EDUCATION

Negros Occidental Branch

Cadiz City

Mathematical Investigation

Submitted to:

Dr. Sandra E. Miranda

Submitted by:Malou A. Caruscay

III-2 BSE-MATH

1stsemester

A.Y 2010-2011

Documents

Euclidian Algorithm Loves One