Find the minimum natural number $n$ so that $131\times n = 123456789\cdots.$ The question is just "Find the minimum natural number $n$ so that $131\times n = 123456789\cdots.$" My first thought was simply just 123456789/131 to get the answer. However, what does that minimum natural number make any difference here? Additionally, dots following the number at the end are really confusing.  I know this might lack sufficient context to get the solution. But really want to discuss here. Any help is welcome! Thank you !
 A: There is probably a deeply number-theoretic approach to this question, but here's a not overly-mathematical solution.
The numbers $123456789...$ are clearly described by tuples of small integers ($n$,$i$) where $n$ is the number of digits and $i$ is the value of the suffix, such that lexicographic ordering of the tuple is identical to the usual order of natural numbers. ie $(0,0) = 123456789$, $(1,0) = 1234567890$, $(1,1) = 1234567891$, $(2,7) = 12345678907$, etc.
So we need to find the smallest such tuple such that the underlying value divides $131$. For the first few $(n,0)\mod 131$ you have (by calculation) $(n,0) = 31,48,87,...$. The value of $(n,i)\mod 131$ is the respective base value of $(n,0)$ plus $i$. The maximum valid value for $i$ for a given $n$ is $0,9,99,...$. The first value of $n$ for which there is a valid value of $i$ such that $(n,0)+i = 0\mod 131$ is clearly $n=2$. By subtraction $i=131-87=44$.
So the smallest such number is $12345678944$.
With the aid of hand-waving and weird diagrams on a blackboard the above answer would be much easier expressed to the point that it is trivial. It's only the limits of notation and brevity that makes it look complex.
A: $123456789/131$ is not an integer, so that's not the answer.  
The dots can stand for any string of digits.
In other words, you want $ 123456789 \times 10^k \le 131 n <   123456790 \times 10^k$ for some nonnegative integer $k$.  For example, $n = 942418237$ is one possible solution, with $k = 3$ and 
$131 n = 123456789047$, but it's not the minimum because there are smaller $n$'s that will work.
A: $$
\frac{123456789101112131415161718192021}{131} = 942418237413069705459249757191.
$$
So there's that.
A: Note that $123456789=131\times 942418\text{ } $  remainder $31$
In the case that you are looking for the minimum when the dots can be any digits, note that with three further digits (therefore $999$ consecutive integers) you can construct a multiple of any positive integer less than $1000$.
If you add one digit you are looking at a remainder between $310$ and $319$. Two digits gives between $3100$ and $3199$, three digits, between $31000$ and $31999$. This defines the search space.
A: The minmal $k$ is $94241824$ , resulting in $12345678944$
