Set closed under quotients and remainders

Question

Suppose that an infinite set $S= {0,1,2,4,8,...}$ of integers written in monotonically increasing order (that is, all other members are integers greater than 8) has the property that Euclidean division of any integer $a$ in $S$ by any integer $b\ne0$ in $S$ (regardless of whether $a>b$) gives quotient $q$ and remainder $r$ also in $S$. That is, $S$ is closed under Euclidean division. Obviously, $S$ could then be ${0,1,2,4,8,16,...}$, but is it possible to prove that there are no other infinite sets $S=0,1,2,4,8,...$ satisfying the closure property?