AIME 1985 Problem 10

How many of the first 1000 positive integers can be expressed in the form $\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$, where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?