Magic Multiple

The Elvish races of Middle Earth believed that certain numbers were more significant than others. When using a particular quantity $n$ of metal to forge a particular sword, they believed that sword would be most powerful if the thickness $k$ were chosen according to the following rule:

Given a nonnegative integer $n$, what is the smallest $k$ such that the decimal representations of the integers in the sequence:

n, 2n, 3n, 4n, 5n, \ldots, kn

contain all ten digits ($0$ through $9$) at least once?

Lord Elrond of Rivendell has commissioned you with the task to develop an algorithm to find the optimal thickness ($k$) for any given quantity of metal ($n$).