1

Let $k$ and $m$ be positive integers. I define a binary operation $k$ rep $m$ as the number you get when writing $k$ $m$ times in base ten. For example, $1$ rep $5$ $=$ $11111$, $23$ rep $2$ $=$ $2323$, and for any positive integer $k$, $k$ rep $1$ $=$ $k$. I conjecture that, for any positive integers $k$, $m$, and $n$, $k$ rep $m$ divides $k$ rep $n$ iff $m$ divides $n$. Is this true, and if so, what is the proof? Of course, one can generalize the operation rep to bases other than ten, such as base two or base three or any other base, and then ask an analogous question. I would like it if the generalized question was answered as well.

user107952
  • 20,508
  • 1
    Special case of the fact that $,f_n = (x^n-1)/(x-1),$ is a strong divisibility sequence so $,f_k\mid f_n\iff k\mid n,\ $ e.g. see here and here in the dupe. – Bill Dubuque Feb 12 '22 at 18:48

1 Answers1

2

Not only is your conjecture evidently always true in every base, the quotient has a simple form in the base used: if $k$ has $d$ digits the quotient is $0\dots01$ rep $n/m$ where there are $d-1$ zeros in the repeating unit (hence $d$ digits in all). For example, in any base with the digits used, $$513513513513513513=000001000001000001×513513$$

Parcly Taxel
  • 103,344