Repdigit ($R_n=d\frac{b^n-1}{b-1}$)

A repdigit is a positive integer whose base-$b$ expansion repeats one nonzero digit $d$. For $b\ge 2$, $1\le d<b$ and $n\ge 1$, the finite geometric series gives $R_n=d(1+b+\cdots +b^{n-1})=d(b^n-1)/(b-1)$. For example, $5(10^n-1)/9$ has $n$ decimal digits, all equal to five.

Every prime number not dividing $b$ divides infinitely many such repdigits. If it divides $d$, every length works. Otherwise, if it does not divide $b-1$, the Fermat little theorem gives divisibility for lengths that are multiples of $p-1$, and division by $b-1$ is justified by its modular inverse. If $p\mid b-1$, the sum is congruent to $dn$ modulo $p$, so lengths divisible by $p$ work instead. A prime dividing $b$ but not $d$ divides none, since $R_n\equiv d(\mathrm{mod}p)$. This distinction prevents invalid cancellation of the geometric-sum denominator.

A repdigit whose digits are all 1 in a specified positional notation of base $b$. Its value is the geometric series $R_b(k)=1+b+\cdots +b^{k-1}=(b^k-1)/(b-1)$. In base ten, appending one digit gives $R_{10}(k+1)=10R_{10}(k)+1$, so its remainder modulo $n$ can be computed by a recurrence relation without constructing the full integer.