Given a pair of integers, $m$ and $n$, is there some $k$ such that $m$ is embedded somewhere in the digits of $ kn $?
By embedded, I mean $m$'s ordered digit sequence is a substring of $kn$. As an example, for $n=1,234$ and $k=1$, the following values of $m$ are embedded in $kn$: $$ m \in \{1,2,3,4,12,23,34,123,234,1234\} $$ If $k=10$, $kn=12,340 $ and possible values of $m$ are $$m \in \{ 1, 2, 3, 4, \mathbf{0}, 12, 23, 34, \mathbf{40}, 123,234,\mathbf{340},1234,\mathbf{2340}, \mathbf{12340}\}$$
If I would like $37$ to be a possible value for $m$ when $n=1,234$ I need only set $k=3$, giving $kn=3,702$ and possible values for $m$ as $$m \in \{3, 7, 0, 2, \mathbf{37}, 70, 370, 702, 3702\}$$
Is there a proof that shows that any $m$ can be found as a substring of some multiple of any $n$? If so, what is the method for finding the smallest value for $k$ that satisfies the embedding requirement, given some pair $(m,n)$?
Note: There may be a better term for "embedded" of which I am unaware. Additionally, I'm not entirely sure under which tags to post this, so if there are other tags it that should be included, I'll make updates.
Sure....pick some huge number $t$ (much larger than either $m,n$) and look at $\alpha= 10^tm$ then there is some multiple of $n$ contained in $\{\alpha ,\alpha +1,\cdots,\alpha+(n-1)\}$ and all of those contain $m$ as the first few digits.
Note: as pointed out in the comments below, taking $t$ to be the length of $n-1$ suffices (so it doesn't have to be all that big).