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.
 A: (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 $5$.
A: 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:
$25$, $5611672278338945$, $229823487399244975$, $\dots$, $1874302285824919569775535$, $1111122222333334444455555666667777788889$
A: Let S be the set of number written with 0 and 1.
Let n be a natural number.
You look for the minimal positive natural number k such  $k \times n \in S$
I will look further to S
Let $s \in S$.
There is a finite subset $J \subset \mathbb{N}$ such:
$s = \sum_{i \ in J} 10^i$
We got:
$k \times n = \sum_{i \ in J} 10^i$
Using mod $n$:
$\sum_{i \ in J} 10^i \equiv 0 \pmod n$
Your problem is now to find $J$
Let $T = \left\{10^i\mod n, i \in \mathbb{N}\right\}$
You now look for a subset of $T$ where the sum of the element is a multiple of $n$.
A greedy algorithm look good here.
A: This is a way to get a multiple containing only $1$ and $0$, it won't be the minimal in some cases, but maybe it can help. You can proceed like this: take $n$ and compute
$$\frac{1}{9n}.$$
Then convert the decimal expansion in the usual fraction, that will be something like
$$\frac{a}{99..0...}.$$
So you obtain
$$\frac{1}{9n}=\frac{a}{99..0...}$$
from which follows
$$a\cdot n =11..00..$$
Example: $n=198$
$$\frac{1}{9\cdot198}=0.000\overline{561167227833894500}=\frac{5611672278338945}{99999999999999990}$$
from which
$$198\cdot5611672278338945=1111111111111111110. $$
