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Ring of triangles II (Posted on 2016-03-10) Difficulty: 4 of 5
As in Ring of triangles I, ABCDEF is a hexagon and O is an interior point. Triangles ABO, CDO, EFO are is equilateral. Their respective side lengths are different integers with no common divisor. (They are also not 4,3,7 because that was part I.)

BC, DE, and FA are also integer lengths.

Find a set of lengths {AB,BC,CD,DE,EF,FA} for which this is possible.

Note: There are sides of the hexagon that can be the same.

No Solution Yet Submitted by Jer    
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Solution re: computer solution -- revised Comment 2 of 2 |
(In reply to computer solution by Charlie)

So that the found sides do not merely reiterate those of the given triangles for any specific set of AB, CD and EF, the following checks that of BC, DE and FA, at least one differs from any in the first (given) set.

But also, I realize I was being too restrictive on the GCD requirement. I had required that no two of AB, CD, EF shared a common factor. The revised program weeds out only those where a common factor is shared among all three of the equilateral triangle side lengths.

2 3 12
2 12 11
41.4096221092709 82.8192442185417 55.7711336721874      180

3 4 7
3 7 6
48.1896851042214 73.3984504009798 58.4118644947988      180

3 9 13
9 11 11
80.4059317731395 56.5449884266241 43.0490798002363      180

3 10 23
8 22 22
41.4096221092709 71.6260619966705 66.9643158940586      180

3 12 23
12 14 22
82.8192442185417 30.2164398873996 66.9643158940586      180

4 7 9
7 8 7
73.3984504009798 58.4118644947988 48.1896851042214      180

5 6 8
5 10 5
53.130102354156 90 36.869897645844      180

5 9 13
10 10 10
86.1774462707257 50.1316584499894 43.690895279285      180

5 9 17
11 13 13
99.5940682268605 48.938955479872 31.4669762932676      180

5 10 26
7 28 23
40.5358021113166 90.881508306806 48.5826895818775      180

5 11 17
9 21 13
53.7784533801649 94.7545703265675 31.4669762932676      180

5 11 20
8 23 17
41.8018441931411 91.0417988504553 47.1563569564037      180

5 12 17
9 21 14
42.8334280660673 91.1235186577004 46.0430532762323      180

5 13 19
10 18 18
43.690895279285 65.3544154841787 70.9546892365363      180

5 15 17
15 18 13
80.4059317731395 68.1270919335929 31.4669762932676      180

6 7 11
7 12 7
64.6230664748477 80.2801322975677 35.0968012275846      180

6 7 22
5 21 20
44.415308597193 72.6203831922193 62.9643082105877      180

6 11 17
13 12 13
95.2159085704541 44.6649245311407 40.1191668984051      180

6 11 18
8 20 15
45.2071662976332 83.4750211558562 51.3178125465106      180

6 13 16
9 11 18
37.3568519729032 43.0490798002363 99.5940682268605      180

6 13 17
11 20 13
57.4210296071959 82.459803494399 40.1191668984051      180

6 13 21
14 14 18
86.6926378203292 40.6905605975357 52.6168015821351      180

6 16 21
14 13 21
60 38.2132107017382 81.7867892982618      180

6 17 19
17 12 17
79.8357513782765 38.4379622530885 61.726286368635      180

7 8 14
9 14 9
73.3984504009798 73.3984504009798 33.2030991980405      180

7 10 25
7 25 22
44.415308597193 78.4630409671845 57.1216504356225      180

7 12 14
11 14 11
64.6230664748477 64.6230664748477 50.7538670503046      180

7 12 26
11 26 21
64.6230664748477 76.6576362029118 38.7192973222405      180

7 13 21
16 14 16
102.051523401152 40.6905605975357 37.2579160013124      180

7 14 23
9 27 20
33.2030991980405 90.3558766951521 56.4410241068074      180

7 15 21
10 18 22
34.0477323699915 57.1216504356225 88.830617194386      180

7 17 27
18 22 22
86.6277133165661 54.4298454144525 38.9424412689814      180

7 18 27
17 27 22
70.5287793655093 70.5287793655093 38.9424412689814      180

7 20 24
15 20 25
36.869897645844 53.130102354156 90      180

7 21 25
22 14 22
88.830617194386 34.0477323699915 57.1216504356225      180

8 9 11
13 5 9
99.5940682268605 26.6274783929746 53.7784533801649      180

8 9 13
11 11 9
80.4059317731395 56.5449884266241 43.0490798002363      180

8 9 14
7 7 18
48.1896851042214 25.2087652967584 106.60154959902      180

8 9 17
11 16 11
80.4059317731395 68.1270919335929 31.4669762932676      180

8 9 17
13 13 11
99.5940682268605 48.938955479872 31.4669762932676      180

8 11 13
9 16 9
53.7784533801649 83.1724668195987 43.0490798002363      180

8 11 13
17 3 9
126.221546619835 10.7293735799286 43.0490798002363      180

8 11 17
9 21 11
53.7784533801649 94.7545703265675 31.4669762932676      180

8 11 19
11 20 13
68.6763137365021 78.6867887599442 32.6368975035537      180

8 11 22
8 22 19
46.5674634422102 75.5224878140701 57.9100487437197      180

8 11 23
5 20 25
24.6199773286571 60.3914806957868 94.9885419755561      180

8 13 17
9 24 11
43.0490798002363 105.483943906496 31.4669762932676      180

8 13 21
6 14 26
18.7169506573713 40.6905605975357 120.592488745093      180

8 15 23
11 26 19
45.5729959991943 83.509211298165 50.9177927026407      180

8 15 25
19 16 19
107.457603123722 37.6271906288337 34.9152062474442      180

8 19 22
13 13 26
32.6368975035537 36.0394162329483 111.323686263498      180

8 19 25
15 33 19
48.8604895851009 96.224304167455 34.9152062474442      180

8 21 27
15 15 31
33.5573097619207 33.5573097619207 112.885380476159      180

9 10 13
5 5 20
29.9264348666142 20.2052235833752 129.868341550011      180

9 10 19
11 11 20
70.5287793655093 26.5253520166097 82.945868617881      180

9 10 20
13 18 13
86.1774462707257 63.8961188626601 29.9264348666142      180

9 10 23
9 23 18
56.2510114041114 77.444142201414 46.3048463944746      180

9 11 13
5 11 17
26.6274783929746 53.7784533801649 99.5940682268605      180

9 11 13
5 18 11
26.6274783929746 96.8275331804013 56.5449884266241      180

9 11 13
8 18 8
45.8156148466955 96.8275331804013 37.3568519729032      180

9 11 13
12 16 6
72.9665898338858 83.1724668195987 23.8609433465154      180

9 11 14
4 11 19
20.0499757241515 50.4788036413578 109.471220634491      180

9 11 16
8 13 17
45.8156148466955 53.7784533801649 80.4059317731395      180

9 11 17
8 8 22
45.8156148466955 22.3114770868973 111.872908066407      180

9 11 19
10 10 22
58.9924169931329 23.9534516247481 97.054131382119      180

9 11 27
5 26 27
26.6274783929746 72.9665898338858 80.4059317731395      180

9 11 27
8 26 24
45.8156148466955 72.9665898338858 61.2177953194186      180

9 11 27
13 17 27
80.4059317731395 19.1881364537209 80.4059317731395      180

9 13 16
11 11 17
56.5449884266241 43.0490798002363 80.4059317731395      180

9 13 17
11 5 22
56.5449884266241 11.5821035069688 111.872908066407      180

9 13 18
10 19 15
50.1316584499894 73.6173301458992 56.2510114041114      180

9 13 24
8 17 27
37.3568519729032 43.0490798002363 99.5940682268605      180

9 13 25
10 18 26
50.1316584499894 43.690895279285 86.1774462707257      180

9 14 18
7 16 21
25.2087652967584 58.4118644947988 96.3793702084428      180

9 16 22
17 18 17
80.4059317731395 53.7784533801649 45.8156148466955      180

9 16 27
17 13 27
80.4059317731395 19.1881364537209 80.4059317731395      180

9 17 24
13 13 27
48.938955479872 31.4669762932676 99.5940682268605      180

9 18 26
15 20 25
56.2510114041114 50.1316584499894 73.6173301458992      180

9 20 22
19 18 19
70.5287793655093 50.4788036413578 58.9924169931329      180

10 11 19
9 24 11
50.4788036413578 102.995844342032 26.5253520166097      180

10 11 23
15 20 15
91.0417988504553 60.3914806957868 28.5667204537579      180

10 13 18
5 19 20
20.2052235833752 73.6173301458992 86.1774462707257      180

10 13 20
9 9 26
43.690895279285 20.2052235833752 116.10388113734      180

10 13 24
13 13 26
67.3801350519596 22.6198649480404 90      180

10 15 23
10 22 22
41.4096221092709 66.9643158940586 71.6260619966705      180

10 17 19
15 20 15
61.1608105993827 67.2322334198243 51.606955980793      180

10 23 27
23 20 23
77.444142201414 46.3048463944746 56.2510114041114      180

11 12 13
19 11 4
111.323686263498 52.020127555148 16.6561861813541      180

11 12 14
7 22 7
35.0968012275846 115.376933525152 29.5262652472631      180

11 12 19
13 11 20
68.6763137365021 32.6368975035537 78.6867887599442      180

11 12 26
17 22 17
95.2159085704541 57.4210296071959 27.36306182235      180

11 13 14
8 13 17
37.7948789632582 57.4210296071959 84.7840914295459      180

11 13 17
6 6 26
27.36306182235 17.3018627087907 135.335075468859      180

11 13 17
11 9 21
53.7784533801649 31.4669762932676 94.7545703265675      180

11 13 19
3 13 27
10.7293735799286 43.0490798002363 126.221546619835      180

11 13 19
4 22 20
16.6561861813541 84.6570250587017 78.6867887599442      180

11 13 19
12 22 12
59.3035587083499 84.6570250587017 36.0394162329483      180

11 13 27
9 35 17
43.0490798002363 117.762783746043 19.1881364537209      180

11 13 27
16 16 26
83.1724668195987 23.8609433465154 72.9665898338858      180

11 14 19
11 9 24
50.4788036413578 26.5253520166097 102.995844342032      180

11 14 21
11 21 18
50.4788036413578 70.5287793655093 58.9924169931329      180

11 14 22
7 24 21
29.5262652472631 80.2801322975677 70.1936024551692      180

11 15 23
10 10 30
41.8018441931411 18.5896365026458 119.608519304213      180

11 15 23
16 22 16
74.1733798681277 66.9643158940586 38.8623042378137      180

11 15 27
6 18 34
20.0499757241515 38.9424412689814 121.007583006867      180

11 16 17
13 9 21
53.7784533801649 31.4669762932676 94.7545703265675      180

11 16 18
13 22 13
53.7784533801649 80.4059317731395 45.8156148466955      180

11 16 24
9 16 29
32.7637577588568 41.4096221092709 105.826620131872      180

11 16 26
13 18 27
53.7784533801649 43.0490798002363 83.1724668195987      180

11 16 27
13 13 32
53.7784533801649 19.1881364537209 107.033410166114      180

11 17 18
8 29 13
22.3114770868973 111.872908066407 45.8156148466955      180

11 17 20
22 17 11
101.411901743813 53.9681209275294 24.6199773286571      180

11 17 26
8 27 27
22.3114770868973 74.5160560935039 83.1724668195987      180

11 17 27
21 11 26
94.7545703265675 12.2788398395466 72.9665898338858      180

11 18 21
11 21 22
35.0968012275846 64.6230664748477 80.2801322975677      180

11 19 21
10 30 18
23.9534516247481 97.054131382119 58.9924169931329      180

11 19 23
18 22 18
67.790739266016 62.3201335598114 49.8891271741725      180

11 19 24
14 7 31
46.8264488927411 13.1735511072589 120      180

11 20 25
13 15 30
36.869897645844 36.869897645844 106.260204708312      180

11 24 27
15 33 26
26.6274783929746 80.4059317731395 72.9665898338858      180

12 13 19
11 24 11
52.020127555148 95.3429749412983 32.6368975035537      180

12 15 20
9 25 16
36.869897645844 90 53.130102354156      180

12 15 23
18 22 14
82.8192442185417 66.9643158940586 30.2164398873996      180

12 15 25
6 26 26
22.3316450092215 76.5851512437694 81.0832037470091      180

12 17 26
11 11 34
40.1191668984051 17.3018627087907 122.578970392804      180

12 19 23
21 22 15
81.9318494999477 62.3201335598114 35.7480169402409      180

12 23 26
29 7 22
107.718931873407 14.8600385193971 57.4210296071959      180

13 14 17
6 24 14
25.331675016713 101.021449635517 53.6468753477701      180

13 14 17
13 11 20
57.4210296071959 40.1191668984051 82.459803494399      180

13 14 22
3 28 21
12.0345697280641 99.7198677024323 68.2455625695036      180

13 14 22
9 28 15
38.7192973222405 99.7198677024323 41.5608349753272      180

13 15 20
4 25 21
14.2500326978036 90 75.7499673021964      180

13 15 20
14 25 11
59.4897625938845 90 30.5102374061155      180

13 15 25
16 26 16
69.2772556184099 76.5851512437694 34.1375931378206      180

13 16 17
11 9 24
43.0490798002363 31.4669762932676 105.483943906496      180

13 16 17
27 9 5
136.950920199764 31.4669762932676 11.5821035069688      180

13 16 18
11 26 11
43.0490798002363 99.5940682268605 37.3568519729032      180

13 16 21
7 7 32
25.331675016713 15.3588855808227 139.309439402464      180

13 16 22
13 22 19
52.020127555148 68.6763137365021 59.3035587083499      180

13 16 27
11 13 35
43.0490798002363 19.1881364537209 117.762783746043      180

13 17 21
14 26 14
53.6468753477701 85.6625640546941 40.6905605975357      180

13 18 20
19 26 9
73.6173301458992 86.1774462707257 20.2052235833752      180

13 18 26
25 20 15
106.382669854101 50.1316584499894 23.4856716959098      180

13 18 27
11 39 16
37.3568519729032 118.782204680581 23.8609433465154      180

13 19 25
10 16 32
29.4895673358721 39.7876882825378 110.72274438159      180

13 19 25
18 26 18
65.3544154841787 70.9546892365363 43.690895279285      180

13 19 27
13 11 35
43.0490798002363 19.1881364537209 117.762783746043      180

13 22 24
19 26 19
59.3035587083499 68.6763137365021 52.020127555148      180

13 23 24
15 13 31
37.7237572831936 32.0239959746187 110.252246742188      180

13 24 27
17 9 35
43.0490798002363 19.1881364537209 117.762783746043      180

14 15 25
13 20 25
53.130102354156 53.130102354156 73.739795291688      180

14 16 19
14 19 17
55.1500954209535 65.098937629633 59.7509669494134      180

14 17 21
13 28 13
48.4046480533817 94.3374359453059 37.2579160013124      180

14 19 21
17 4 29
59.7509669494134 9.94884220867952 110.300190841907      180

14 21 23
17 28 17
53.7125429148901 78.8985626384974 47.3888944466125      180

14 21 26
13 17 32
37.2579160013124 40.6905605975357 102.051523401152      180

14 21 27
15 18 31
45.5183921612098 41.7522052020115 92.7294026367787      180

14 22 25
20 7 31
62.9643082105877 15.4987327565968 101.536959032815      180

14 24 27
18 21 29
48.1896851042214 48.1896851042214 83.6206297915572      180

15 16 23
11 11 32
41.4096221092709 25.5546937847878 113.035684105941      180

15 16 25
13 13 32
49.4583981264955 27.126753117274 103.414848756231      180

15 17 19
10 30 10
35.7312919799637 112.767766580176 31.5009414398606      180

15 17 27
18 11 33
68.1270919335929 12.2788398395466 99.5940682268605      180

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22 22 18
86.6277133165661 54.4298454144525 38.9424412689814      180

15 18 20
9 26 19
29.9264348666142 86.1774462707257 63.8961188626601      180

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27 8 17
109.471220634491 20.0499757241515 50.4788036413578      180

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9 40 13
29.9264348666142 129.868341550011 20.2052235833752      180

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29 7 17
116.574969518987 15.7948290503887 47.6302014306242      180

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15.2227570341893 70.9546892365363 93.8225537292743      180

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8 8 38
23.6819072911771 13.9452833376567 142.372809371166      180

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17 19 25
58.5971135386212 48.8604895851009 72.5423968762779      180

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25 20 15
90 53.130102354156 36.869897645844      180

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45.5729959991943 50.9177927026407 83.509211298165      180

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17 35 18
50.4788036413578 90.5787550896608 38.9424412689814      180

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21 26 7
78.9785503644831 85.6625640546941 15.3588855808227      180

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9 9 34
31.4669762932676 22.3114770868973 126.221546619835      180

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3 23 32
9.83822698524653 65.6842608288235 104.47751218593      180

16 17 27
9 26 29
31.4669762932676 68.1270919335929 80.4059317731395      180

16 17 27
9 40 13
31.4669762932676 129.344887253012 19.1881364537209      180

16 19 27
10 34 22
31.7525344485699 93.6882400794739 54.5592254719562      180

16 19 27
17 32 17
57.35218256486 86.3117599205261 36.3360575146139      180

16 23 24
13 17 32
33.1798823618629 42.3426054522072 104.47751218593      180

16 23 25
15 18 31
40.4590830807894 43.8017464419438 95.7391704772668      180

16 24 27
15 18 34
37.7804045666502 40.8044376906193 101.41515774273      180

17 18 20
5 22 27
16.0989339510568 70.5287793655093 93.3722866834339      180

17 18 22
29 16 9
111.872908066407 45.8156148466955 22.3114770868973      180

17 18 24
10 39 8
33.045762472559 135.951374325916 11.0028632015253      180

17 19 20
18 26 12
59.6922793231181 83.5784842199182 36.7292364569637      180

17 19 24
3 17 37
7.13324087398138 44.7222460247886 128.14451310123      180

17 19 24
9 23 29
28.2603825592097 63.425030481013 88.3145869597774      180

17 19 27
16 34 16
52.4172125069794 93.6882400794739 33.8945474135467      180

17 20 22
27 18 13
93.3722866834339 50.4788036413578 36.1489096752083      180

17 20 25
17 25 22
53.9681209275294 66.4218215217982 59.6100575506724      180

17 21 23
14 34 14
41.5912771433651 101.101437361503 37.3072854951323      180

17 21 23
32 16 10
114.315739171176 42.3426054522072 23.3416553766163      180

17 21 24
26 30 8
85.6625640546941 83.3345727437806 11.0028632015253      180

17 22 26
9 32 27
22.3114770868973 83.1724668195987 74.5160560935039      180

17 23 24
16 19 29
44.0552116095984 47.6302014306242 88.3145869597774      180

17 24 27
13 9 40
31.4669762932676 19.1881364537209 129.344887253012      180

17 24 27
13 33 26
31.4669762932676 80.4059317731395 68.1270919335929      180

18 19 23
11 36 11
34.4559504612953 117.679866440189 27.8641830985161      180

18 19 25
13 36 13
41.0282543699221 109.045310763464 29.9264348666142      180

18 19 27
2 34 30
5.36820469916192 93.6882400794739 80.9435552213642      180

18 20 27
22 17 27
70.5287793655093 38.9424412689814 70.5287793655093      180

18 22 23
32 3 23
105.826620131872 7.20906397406899 66.9643158940586      180

18 22 27
12 23 33
33.0301516046367 54.84729703322 92.1225513621434      180

18 25 26
13 9 40
29.9264348666142 20.2052235833752 129.868341550011      180

18 26 27
12 17 39
23.8609433465154 37.3568519729032 118.782204680581      180

18 26 27
16 11 39
37.3568519729032 23.8609433465154 118.782204680581      180

19 20 22
9 38 9
26.5253520166097 129.521196358642 23.9534516247481      180

19 21 22
12 27 23
34.4559504612953 77.7533102726887 67.790739266016      180

19 21 24
16 39 7
46.8264488927411 120 13.1735511072589      180

19 22 24
13 38 13
36.0394162329483 111.323686263498 32.6368975035537      180

19 23 25
10 22 34
25.3257813471175 54.3657656497948 100.308453003088      180

19 23 26
6 42 20
12.2808543958512 117.865093266277 49.8540523378715      180

19 23 26
12 23 33
31.4004274006256 55.5826112895584 93.016961309816      180

19 25 27
26 38 10
70.9546892365363 93.8225537292743 15.2227570341893      180

20 22 25
22 7 35
62.9643082105877 15.4987327565968 101.536959032815      180

20 22 27
18 35 17
50.4788036413578 90.5787550896608 38.9424412689814      180

20 22 27
38 7 17
129.521196358642 11.5363623723765 38.9424412689814      180

20 23 27
7 14 43
16.9574262941723 31.2322588100491 131.810314895779      180

21 22 23
29 9 26
84.7840914295459 22.9348404438612 72.2810681265929      180

21 22 25
7 7 44
18.5489996133947 15.4987327565968 145.952267630008      180

21 22 26
29 16 23
84.7840914295459 37.7948789632582 57.4210296071959      180

21 24 25
7 20 39
16.1951167389636 48.1255944005544 115.679288619457      179.999999758975

21 25 27
31 13 27
84.2608295227332 28.6245509534254 67.1146195238414      180

22 24 25
10 41 17
24.6199773286571 113.578178478202 41.8018441931411      180

23 25 26
42 21 5
122.034836044198 48.5826895818775 9.38247437392496      180

23 25 27
16 4 46
38.6597389359259 7.64510745854866 133.695153605525      180

23 25 27
18 47 5
43.8017464419438 129.296480239188 6.90177331886874      180

DefDbl A-Z
Dim crlf$, pi


Private Sub Form_Load()
 Form1.Visible = True
 
 crlf = Chr(13) + Chr(10)
 Text1.Text = ""
 
 pi = 4 * Atn(1)
 
 For ab = 1 To 25
 For cd = ab + 1 To 26
 For ef = cd + 1 To 27
   DoEvents
   If gcd(gcd(ab, cd), ef) = 1 Then
    For bc = cd - ab + 1 To cd + ab - 1
    For de = ef - cd + 1 To ef + cd - 1
    For fa = ef - ab + 1 To ef + ab - 1
      found = 0
      If bc <> ab And bc <> cd And bc <> ef Then found = 1
      If de <> ab And de <> cd And de <> ef Then found = 1
      If fa <> ab And fa <> cd And fa <> ef Then found = 1
      If found Then
       a1 = angle(bc, cd, ab)
       a2 = angle(de, ef, cd)
       a3 = angle(fa, ef, ab)
       diff = Abs(180 - a1 - a2 - a3)
       If diff < 0.000001 Then
         Text1.Text = Text1.Text & ab & Str(cd) & Str(ef) & crlf
         Text1.Text = Text1.Text & bc & Str(de) & Str(fa) & crlf
         Text1.Text = Text1.Text & a1 & Str(a2) & Str(a3) & "     " & Str(a1 + a2 + a3) & crlf & crlf
       End If
      End If
    Next
    Next
    Next
   End If
 Next
 Next
 Next

 Text1.Text = Text1.Text & crlf & " done"
  
  
End Sub

Function angle(a, b, c)
  cosa = (b * b + c * c - a * a) / (2 * b * c)
  If cosa = 0 Then angle = 90: Exit Function
  sina = Sqr(1 - cosa * cosa)
  a1 = Atn(sina / cosa) * 180 / pi
  If cosa < 0 Then a1 = a1 + 180
  angle = a1
End Function

Function gcd(a, b)
  x = a: y = b
  Do
   q = Int(x / y)
   z = x - q * y
   x = y: y = z
  Loop Until z = 0
  gcd = x
End Function


  Posted by Charlie on 2016-03-10 10:09:56
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