Think of this problem as an extension to
Repeating decimals.
Determine the minimum value of base 14 positive integer q such that the tetradecimal representation of 1/q has a repetition length of n for each of {1, 2, ..., 9, A, B, C, D, 10}
Is there a simple way of finding such a number?
Each section below is headed by the base being used. After that the column heads apply.
length q repeated
of (decimal) digits
cycle
8
1 7 1
2 3 25
3 73 007
4 5 1463
5 31 02041
6 19 032745
7 49 0123457
8 17 03607417
9 262657 000000777
10 11 0564272135
11 23 02620544131
12 37 015654762123
13 79 0063662164521
14 43 01372027640575
15 631 000637342304271
16 97 0052164077256137
9
1 2 4
2 5 17
3 7 125
4 32 0247
5 11 07324
6 35 022741
7 547 0012885
8 17 04678421
9 19 042327518
10 25 0321385675
11 23 03462311507
12 224 003225447671
13 398581 0000012888885
14 29 02712148617674
15 31 025457228114052
16 128 0056227845116734
10
1 3 3
2 11 09
3 27 037
4 101 0099
5 41 02439
6 7 142857
7 239 0041841
8 73 01369863
9 81 012345679
10 451 0022172949
11 21649 00004619151
12 707 001414427157
13 53 0188679245283
14 2629 00038037276531
15 31 032258064516129
16 17 0588235294117647
11
1 2 5
2 3 37
3 7 163
4 16 0762
5 25 04927
6 9 124986
7 43 028A541
8 32 03865931
9 1772893 000000AAA
10 75 0168238A46
11 15797 0000A21665A
12 13 093425A17685
13 1093 0012439082128
14 129 00A3550382A914
15 175 007673229895451
16 17 07132651A3978459
12
1 11 1
2 13 0B
3 157 00B
4 5 2497
5 22621 0000B
6 7 186A35
7 659 0027571
8 89 0174BA47
9 37 03A85232B
10 19141 00010BBBAB
11 23 06316948421
12 35 0414559B3931
13 477517 000006305530B
14 211 0082336BB39885
15 61 0243B2683192B4B
16 17 08579214B36429A7
13
1 2 6
2 7 1B
3 9 15A
4 5 27A5
5 30941 0000C
6 63 028B47
7 5229043 000000C
8 32 05386BA2
9 27 0634A791C
10 11 12495BA837
11 23 0746A235119
12 45 039A8C564812
13 53 0325B6B38AA3C
14 29 05A9B28C7231A4
15 4651 00061AA53222739
16 64 028435B7691A9C51
14
1 13 1
2 3 49
3 211 00D
4 197 00DD
5 11 13B65
6 9 17AC63
7 8108731 000000D
8 41 04ACD931
9 397 006CAA12B
10 25 07BA8D6235
11 67 02CD531C485
12 37 05423AD89BA3
13 157 013698B8415D9
14 7027567 0000010DDDDDCD
15 31 0647323881B40C9
16 17 0B75A9C4D2683419
15
1 2 7
2 4 3B
3 241 00E
4 64 037B
5 11 156C4
6 211 010EDE
7 49 048D26B
8 17 0D37E1B7
9 541 006389B5E
10 31 073D0E7B1E
11 67 03558E4DD32
12 13 124936DCA5B8
13 53 043A2C6BD47DD
14 196 0123456789ABCE
15 61 03A4DB856232E3E
16 256 00D2B496785A3C1E
16
1 3 5
2 17 0F
3 7 249
4 257 00FF
5 11 1745D
6 119 0226B9
7 29 08D3DCB
8 65537 0000FFFF
9 19 0D79435E5
10 187 015E75BB8D
11 23 0B21642C859
12 97 02A3A0FD5C5F
13 53 04D4873ECADE3
14 493 0084EEDD357C1B
15 61 04325C53EF368EB
16 641 00663D80FF99C27F
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Posted by Charlie
on 2022-10-25 13:59:50 |
for bases 8 through 16
