Find the smallest value of $n$ such that $19 \,|\, 10^n+1$ A riddle I'm working on reduces to this question. I don't have a number theory background and don't really know how to approach this kind of problem.
In fact I'm not sure such an $n$ exists. I feel like it does, but my computer has been chugging on an R script for almost 20 minutes to find one.
Any suggestions for resources/additional reading about evaluating divisibility appreciated!

The pattern
$\begin{array}{l}
10^3 + 1 = 1001 = 999+2 \\
10^4 + 1 = 10001 = 9999+2\\
10^5 + 1 = 100001 = 99999+2\\
\dots \\
10^9 + 1 = 999999999+2
\end{array}$
Multiples of $19$:
\begin{array}{r|ccccccccc}
     n &  1 &  2 &  3 &  4 &  5 &   6 &   7 &   8 &   9 \\
   19n & 19 & 38 & 57 & 76 & 95 & 114 & 133 & 152 & 171 \\
\end{array}
$$\begin{array}{cccccccccc}
&  &   & 5 & 2 & 6 & 3 & 1 & 5 & 7 & 9\\  
&  & - & - & - & - & - & - & - & - & -\\
19 & ) & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 \\
& & 9 & 5 & &  \\  
& & - & -\\
& & & 4 & 9\\
& & &  3 & 8 \\
& & & - & -\\
& & & 1 & 1 & 9 \\
& & & 1 & 1 & 4 \\
& & & - & - & -\\
& & & &   & 5 & 9 \\
& & & &   & 5 & 7 \\
& & & & & & - & - \\
& & & & &   & 2 & 9\\
& & & & &   & 1 & 9\\
& & & & & & - & - \\
& & & & &   & 1 & 0 & 9\\
& & & & &   &   & 9 & 5\\
& & & & & & & - & - & -\\
& & & &   & &   & 1 & 4 & 9\\
& & & & &   &   & 1 & 3 & 3\\
& & & & &   &   & - & - & -\\
& & & & &   &   &   & 1 & 6 & 9\\
& & & & &   &   &   & 1 & 7 & 1\\
& & & & &   &   &   & - & - & -\\
& & & & &   &   &   &   &   & -2\\
\end{array}
$$
Hence 
$999999999 = 19 \times 52631579 - 2$
$10^9 + 1 = 19 \times 52631579$
$19 \mid 10^9+1$
