1

Let $R_n$ denote the repunit with $n$ ones. So $R_1=1, R_2=11, R_3=111, ...$.

I know that $R_n = (10^n - 1)/9$ and I want to show that

$R_n$ divides $R_m$ iff $n$ divides $m$.

How would I go about doing that? I tried to use the definition of divisibility but I end up with $log_{10}$ terms and there wasn’t much I could do after that.

Bill Dubuque
  • 272,048
Math55
  • 143
  • As a first step : $R_n\mid R_m$ is equivalent to $10^n-1\mid 10^m-1$ – Peter Feb 12 '23 at 14:37
  • The $10$ is not important, any base works the same. You can see any of these duplicates and the ones they reference: https://math.stackexchange.com/questions/3723091/2r-1-divides-2rs-1, https://math.stackexchange.com/questions/1247197/a-different-way-to-show-that-ak-1-mid-an-1-if-k-mid-n, https://math.stackexchange.com/questions/315210/divisibility-problem-number-theory?noredirect=1, 2p, https://math.stackexchange.com/questions/223818/2m-12n-1-divides-2mn-1-if-and-only-if-gcdm-n-1?noredirect=1 – Ross Millikan Feb 12 '23 at 14:47

0 Answers0