Why $142857$ when multiplied by $1$ to $6$ gives the same digits?

Question

What made 142857 a special number? Why it gives the same digits if it is multiplied by 1,2,3,4,5 & 6 ? And gives all nines when it is multiplied by 7?

It just got struck in my head... And it started as a puzzle to me — user698179, Aug 24 '19 at 18:30

score 9 · Accepted Answer · answered Aug 24 '19 at 18:42

9

More generally, this happens for the fraction $1/n$ exactly when $10$ is a primitive root mod $n$.

Those $n$ are the ones in A167797: $$ 7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, \dots $$

answered Aug 24 '19 at 18:42

lhf

216,483

See also https://en.wikipedia.org/wiki/Full_reptend_prime – lhf Aug 24 '19 at 18:49
See also https://en.wikipedia.org/wiki/Cyclic_number – lhf Aug 25 '19 at 01:37

score 3 · Answer 2 · answered Aug 24 '19 at 18:28

3

Because $\dfrac1{7} =.142857142857... $ and all (and there is a lot) that follows from that.

answered Aug 24 '19 at 18:28

marty cohen

107,799

1

I don't think just mentioning this is helpful at all. The OP might have observed this besides you haven't mentioned how to further draw conclusions from this. – Aug 29 '19 at 19:52
Thank you. Upvoted. – marty cohen Aug 29 '19 at 23:11

Why $142857$ when multiplied by $1$ to $6$ gives the same digits?

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