Does there exist $n$ such that all numbers $n,2n,\dots,2000n$ have the same digits? 
Does there exist a number $n$ such that all numbers $n, 2n, 3n, 4n, \dots, 2000n$ have the same multi-sets of digits except zeroes?

(Having the same multi-sets of digits excepts zeroes means having equal number of ones, twos, ... , nines in the decimal expansion.)
A related question was already asked on MathSE, but the answers there does not provide an approach suitable for bigger numbers like 2000. 
 A: Suppose $N>2000$ is an integer such that the period length of the (eventually) repeating $\frac1N$ equals $N$. Then in computing the decimal expansion all remainders $1,\ldots,N-1$ occur at some place. Then the fractions $\frac1N,\frac2N,\ldots, \frac{2000}N$ turn out to lead to the very same period, merely shifted.
In this situation,  we have $\frac1N=\frac n{10^{N-1}-1}$ for some $1\le n< 10^N-1$ and conclude that $n,2n,\ldots,2000n$ indeed are obtained by rotating the digit sequences suitably (taking an appropriate number of leading zeroes into account).
(Actually, it would be sufficient that the remainders $1,2,\ldots,2000$ occur during the computation of the period, so the period length might be smaller than $N$.)
The question is: Do such $N$ exist?
Primes are good candidates (for any other $N$, the order of $10$ cannot exceed $\phi(N)$). So for which primes $N$ is $10$ of order $N-1$? One such prime is $2017$ and that is $>2000$, thius solving the concrete problem - or rather
$$ n=\frac{10^{2016}-1}{2017}=\underbrace{4957858205\ldots233019335647}_{2013\text{ digits}}
$$
is. Intriguingly, each of the digits $1,2,4,5,7,8$ occurs $202$ times, each of $3,6,9$ occurs $201$ times in that $n$.
In fact, it turns out that $2017$ is the smallest prime $>2000$ with this property. As additionally, $\phi(n)<2000$ for all composites $<2017$, we see that $2017$ is the smallest $N$ for which the above construction works. However, this does not completely rule out that smaller $n$ exist (where  the $kn$ only "accidentally" have the same digit statistics).
See also sequence Full reptend primes in OEIS.
A: I told my friend Ruby how to check the numbers by Hagen, zhoraster and two test numbers by me (n=4212345, n=0). She did the check and got:
n digits:
[0:198, 1:202, 2:202, 3:201, 4:202, 5:202, 6:201, 7:202, 8:202, 9:201]
OK - congrats!

n digits:
[0:227, 1:231, 2:231, 3:231, 4:231, 5:231, 6:231, 7:231, 8:231, 9:230]
OK - congrats!

n digits:
[0:0, 1:1, 2:2, 3:1, 4:2, 5:1, 6:0, 7:0, 8:0, 9:0]
error:
n_2 digits:
[0:1, 1:0, 2:1, 3:0, 4:2, 5:0, 6:1, 7:0, 8:1, 9:1]

n digits:
[0:1, 1:0, 2:0, 3:0, 4:0, 5:0, 6:0, 7:0, 8:0, 9:0]
OK - congrats!

We then did a little search for numbers of the form $(10^{k-1}-1)/k$ for $k$ from $1$ to $3000$. See the log.
