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.

Let $mj = n$, where $m$ is a number from your list, $j$ is some unknown multiplier, and $n$ is the resulting product consisting entirely of $1$'s and $0$'s.
You can solve this with a dynamic programming / breadth-first search approach. The important thing to note is that we don't actually need to look directly for $j$, but rather the smallest (valid) $n$ such that $n \equiv 0 \bmod m$ (at which point we can do $n/m$ to get $j$). 
By traversing through the digits of $n$ (where each digit is either a $1$ or a $0$), we can keep track of the residues and stop early once we reach a mod-$0$ case, assuming lexicographically-minimal traversal.
This approach yields answers to each number in your list almost instantly:
$25$, $5611672278338945$, $229823487399244975$, $\dots$, $1874302285824919569775535$, $1111122222333334444455555666667777788889$
