Solution:

Suppose $k$ is not an integer. Take an integer $n$ such that $nk > 1$, but $nk$ is not an integer. Now take a positive integer $c$ such that $\frac{1}{c+1} \leq nk - [nk] < \frac{1}{c}$. Then $1 \leq (c+1)nk - (c+1)[nk] < 1 + \frac{1}{c}$. Hence $[(c+1) n k] = (c+1)[n k] + 1$. Put $m = (c+1) n$. Then $m$ is a multiple of $n$. But if $[mk]$ is a multiple of $[nk]$, then $[mk] - (c+1)[nk] = 1$ is a multiple of $[nk]$, which is impossible since $nk > 1$. So we have a contradiction. So $k$ must be an integer.