From Newsgroup: rec.puzzles

Chat GPT came up with an interesting question. I have modified it slightly
to involve some different options. I believe I have solutions, for each
part, derived programmatically. ChatGPT suggests a more sophisticated
solution for part f).

Questions:

a) Find the smallest positive integer n, such that 2n is a digit-
permutations of n?

b) Find the smallest positive integer n, such that 3n, is a digit-
permutations of n?

c) Find the smallest positive integer n, such that 4n is a digit-
permutations of n?

d) Find the smallest positive integer n, such that 5n is a digit-
permutations of n?

e) Find the smallest positive integer n, such that 6n is a digit-
permutations of n?

f) Find the smallest positive integer n, such that 2n, 3n, 4n, 5n, and 6n
are all digit-permutations of n?
--
David Entwistle
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From Newsgroup: rec.puzzles


David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

Chat GPT came up with an interesting question. I have modified it slightly to involve some different options. I believe I have solutions, for each part, derived programmatically. ChatGPT suggests a more sophisticated solution for part f).

Questions:

a) Find the smallest positive integer n, such that 2n is a digit- permutations of n?

b) Find the smallest positive integer n, such that 3n, is a digit- permutations of n?

c) Find the smallest positive integer n, such that 4n is a digit- permutations of n?

d) Find the smallest positive integer n, such that 5n is a digit- permutations of n?

e) Find the smallest positive integer n, such that 6n is a digit- permutations of n?

f) Find the smallest positive integer n, such that 2n, 3n, 4n, 5n, and 6n are all digit-permutations of n?

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2 * 125874 = 251748

3 * 1035 = 3105

4 * 1782 = 7128

5 * 142857 = 714285

6 * 1386 = 8316

2 * 142857 = 285714
3 * 142857 = 428571
4 * 142857 = 571428
5 * 142857 = 714285
6 * 142857 = 857142
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From Newsgroup: rec.puzzles

On Wed, 20 May 2026 13:31:08 GMT, Ilan Mayer wrote:

SPOILER

Well done.

ChapGPT provided the following commentary on the solution of part f),
which seems illuminating.

Rnpu cebqhpg vf n crezhgngvba (n plpyvp ebgngvba) bs 142857, fb gur
pbaqvgvba ubyqf.

Jul guvf vf gur fznyyrfg (fxrgpu):

Vs 2a vf n crezhgngvba bs a gura 2a rei a (zbq 9), fb a rei 0 (zbq 9) rCo a
vf qvivfvoyr ol 9.
Gur jryy-xabja ercrngvat qrpvzny sbe 1/7 = 0.142857142857... cebqhprf
gur 6-qvtvg plpyvp ahzore 142857 jubfr zhygvcyrf 2rCo6 ner vgf plpyvp ebgngvbaf; guvf tvirf n inyvq rknzcyr.
Bar pna purpx gung ab fznyyre ahzore (jvgu srjre guna 6 qvtvgf) pna fngvfsl gur pbafgenvagf (qvtvg-fhz naq qvtvg-pbhag pneel pbafgenvagf sbepr
ng yrnfg 6 qvtvgf), fb 142857 vf gur zvavzny fbyhgvba.
--
David Entwistle
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