Question 7
There is a set of $1000$ switches, each of which has four positions, called $A,$ $B,$ $C,$ and $D.$ When the position of any switch changes, it is only from $A$ to $B,$ from $B$ to $C,$ from $C$ to $D,$ or from $D$ to $A.$ Initially each switch is in position $A.$ The switches are labeled with the $1000$ different integers $2^x 3^y 5^z,$ where $x,$ $y,$ and $z$ take on the values $0,$ $1,$ $\ldots,$ $9.$ At step $i$ of a $1000$-step process, the $i$th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$th switch. After step $1000$ has been completed, how many switches will be in position $A?$