Talk:Minimum multiple of m where digital sum equals m: Difference between revisions
Talk:Minimum multiple of m where digital sum equals m (view source)
Revision as of 11:02, 8 February 2022
, 2 years agosome preliminary brute force attack results
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(some preliminary brute force attack results) |
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:Very good. That trailing zero optimisation is just ''amazing''. I used a custom counting method instead of combinatorics but it's pretty much the same idea really. 165 appears to be quite knarly, or at least around 100s which is 99 more than the total time taken for all of 1..164. 275 is even worse. --[[User:Petelomax|Pete Lomax]] ([[User talk:Petelomax|talk]]) 23:22, 5 February 2022 (UTC)
::I must find the time to code this! Until such time I need to add a new rule when 11 is a prime factor. For a number to be divisible by 11 the sum of the digits in the odd places - the sum of the digits in the even places must be a multiple of 11 including zero. Candidates should be constructed with this in mind. Note that the above rules can of course be extended to include 200, 100, 50, and 25--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 12:01, 7 February 2022 (UTC)
:::Never heard of that rule for numbers divisible by 11 before, certainly sounds like it might be be rather handy.<br>
:::Here are some preliminary results of a brute force attack (wading through as many billion numbers as it could in a few seconds or so)<br>
:::Actual results are marked with an asterisk, obviously I've carried on regardless looking for and/or proving any patterns.<br>
:::Constructing the first candidates might be a tad tricker than it first looks... --[[User:Petelomax|Pete Lomax]] ([[User talk:Petelomax|talk]]) 11:00, 8 February 2022 (UTC)
<pre>
For n=11:
We can discount all 8 potential 2-digit candidates 29,38,47,56,65,74,83,92 as none are divisible by 11.
There are only 8 potential 3-digit candidates: 209*,308,407,506,605,704,803,902, and
there are only 8 potential 4-digit candidates: 2090,3080,4070,5060,6050,7040,8030,9020.
There are only 61 potential 5-digit candidates: 10109,10208,10307,10406,10505,10604,10703,10802,10901,20009,..90200, again
there are only 61 potential 6-digit candidates, the same set as 5-digits but with a trailing zero (but not so for n>11).
there are only 279 potential 7-digit candidates: 1000109..9020000, "" for 8-digit candidates
there are only 992 potential 9-digit candidates: 100000109..902000000, "" for 10-digit candidates
The first potential 11 digit candidate is 10000000109
For n=22 (but including all the odd candidates for now):
There are no potential 2 or 3 digit candidates at all.
There are only 64 potential 4 digit candidates: 2299,2398*,..2992,3289,..9724,9823,9922.
There are 509 potential 5 digit candidates: 12199,12298,12397,..12991,13189,..98230,99022,99121,99220.
There are 4230 potential 6 digit candidates: 101299..992200
There are 19770 potential 7 digit candidates: 1002199..9922000
There are 97611 potential 8 digit candidates: 10001299..99220000
There are 350676 potential 9 digit candidates: 100002199..992200000
There are 1334740 potential 10 digit candidates: 1000001299..9922000000
The first potential 11 digit candidate is 10000002199
For n=33 (and dropping the limit by 1 digit as it is starting to crawl):
There are no potential 2, 3, or 4 digit candidates at all.
There are only 168 potential 5 digit candidates: 42999*,43989,44979,45969,..99726,99825,99924.
There are 2730 potential 6 digit candidates: 141999..999240
There are 41690 potential 7 digit candidates: 1032999..9992400
There are 331292 potential 8 digit candidates: 10041999..99924000
There are 2437485 potential 9 digit candidates: 100032999..999240000
The first potential 10 digit candidate is 1000041999
for n=44:
There are no potential 2..5 digit candidates at all.
There are 441 potential 6 digit candidates: 449999..479996*..999944
There are 12279 potential 7 digit candidates: 1439999..9999440
There are 292800 potential 8 digit candidates: 10349999..99994400
There are 3568545 potential 9 digit candidates: 100439999..999944000
The first potential 10 digit candidate is 1000349999
for n=55:
There are no potential 2..6 digit candidates at all.
There are 420 potential 7 digit candidates: 6499999..9999946
There are 21180 potential 8 digit candidates: 16399999..60989995*..99999460
There are 1042440 potential 9 digit candidates: 105499999..999994600
The first potential 10 digit candidate is 1006399999
for n=66 (and we can just afford to increase the limit again):
There are no potential 2..7 digit candidates at all.
There are 400 potential 8 digit candidates: 66999999..67999998*..99999966
There are 37200 potential 9 digit candidates: 165999999..999999660
There are 3067425 potential 9 digit candidates: 1056999999..9999996600
The first potential 11 digit candidate is 10065999999
for n=77:
There are no potential 2..8 digit candidates at all.
There are 100 potential 9 digit candidates: 869999999..899897999*..999999968
There are 17350 potential 10 digit candidates: 1859999999..9999999680
The first potential 11 digit candidate is 10769999999
for n=88 (ditto):
There are no potential 2..9 digit candidates at all.
There are 25 potential 10 digit candidates: 8899999999..9999999988
There are 14960 potential 11 digit candidates: 18799999999..19999999888*..99999999880
The first potential 12 digit candidate is 107899999999
for n=99:
There are no potential 2..12 digit candidates at all (is that right?).
The first potential 13 digit candidate is 1098999999999*
for n=110 (without the trailing zero optimisation):
There are no potential 2..13 digit candidates at all.
The first potential 14 digit candidate is 11999999999999 (119999999999990*, 15 digits)
</pre>
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