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?

Answer 1

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: $$ 7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, \dots $$

Answer 2

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