euclidian algorithm loves one
TRANSCRIPT
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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
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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+
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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
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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
-is the remainder of step 2
3. The GCDs of the two consecutive even numbers and two reversed 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
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
-is the remainder of step 1
-is the remainder of step 2
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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
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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
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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
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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
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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:
- E is the initial even number
-e is the preceded even number
-O is the initial odd number
-o is the preceded odd number
Please refer to DATA 1
Let 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
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
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
http://en.wikipedia.org/wiki/Greatest_common_divisorhttp://en.wikipedia.org/wiki/Greatest_common_divisor -
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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