Repunit

In a positional notation of integer radix $b\ge 2$, a repunit is a repdigit whose numerical digits are all $1$. A length-$k$ repunit, with $k\ge 1$, is the geometric seriesFor example, $R_{10}(4)=1111$, while $R_2(4)=(1111{)}_2=15$. The same string has different values in different radices.

Appending one numerical digit gives $R_b(k+1)=b{R}_b(k)+1$. Thus remainders modulo a positive integer $n$ obey $r_{k+1}=(b{r}_k+1)\mathrm{mod}n$, starting with $r_0=0$. This recurrence relation avoids constructing huge integers. For decimal repunits modulo $7$, the successive remainders are $1,4,6,5,2,0$; hence the six-digit repunit is divisible by $7$.

A repunit divisible by $n$ exists exactly when $b,n$ are coprime integers, and one then exists with at most $n$ digits. Necessity follows because a common prime factor of $b,n$ divides every nonconstant term of $R_b(k)$, leaving remainder one. For sufficiency, examine $R_b(1),\dots ,R_b(n)$ modulo $n$. If none is zero, the pigeonhole principle gives equal remainders at two indices $i>j$. Their difference is $b^j{R}_b(i-j)$. Cancelling the invertible factor $b^j$ modulo $n$ produces a divisible repunit of length $i-j\le n$.