What's the smallest number that we can multiply with a given one to get the result only zeros and ones? I have the following set of numbers,
$$4, 198, 4356, 10296, 14454, 25542, 31779, 51252, 53946, 99999$$
Let's take $3,4$ as an examples:
The smallest number to multiply with $4$ to get the result only $1$s and $0$s is
$$100 = 4 * 25$$
so the result is $25$ for input $4$.
Also
$$111 = 3 * 37$$
so the result is $37$ for input $3$.
Hint:
As you may notice that the rest of the numbers can be divided by $9$ or $99$ or $999$ and the smallest number to multiply with $9$ is $12345679$, this might be the trick!
$$111111111 = 9 * 12345679$$
And one last thing: This is not a binary conversion as most people think by looking at the 4 example.

(Not an answer; a bit too long for a comment.)
I don't know if it is clear, looking at the original question, that such a number exists for any given input $n \in \mathbb{N}$. And so here is a quick proof of that fact, which shows that the question makes sense:
Consider the numbers $1, 11, 111, \ldots$ and observe that for an input $n$, we can reduce each of the numbers in that sequence modulo $n$; let us do so for the first $n$ elements of the sequence. If any returns $0$, then we have found a number with the desired property. If none returns $0$, then we have $n$ returns from a set of $n-1$ residues; by the pigeonhole principle, some residue repeats, and the difference of two $1\ldots1$ numbers that are equivalent modulo $n$ will have the desired property. 
Note: This approach always produces an output consisting of concatenated $1$s followed by concatenated $0$s; the $0$s are necessary if and only if $n$ is divisible by $2$ or $5$.
