Start with the fraction \({n \over 3}\). This is only terminating when n is a multiple of 3.
Now, look at the \({n \over 1375}\) fraction. This is only terminating when n is a multiple of 11.
So, every \(\text{lcm}(3,11) = 33\) integers, n will terminate.
Thus, there are \(\lfloor {1000 \over 33} \rfloor = \color{brown}\boxed{30}\) integers that work.