We consider an unstable scalar linear stochastic system, $X_{i+1}=a X_i+U_i + Z_i$, where $a geq 1$ is the system gain, $Z_i$'s are random variables with bounded $alpha$-th moments, and $U_i$'s are the control actions that are chosen by a controller who receives a single element of a finite set ${1, ldots, M}$ as its only information about system state $X_i$. We show that $M = lfloor a + 1 rfloor$ is necessary and sufficient for $beta$-moment stability, for any $beta < alpha$. The converse is shown using information-theoretic techniques.  The matching achievability scheme is a  uniform quantizer of zoom-in / zoom-out type whose performance is analyzed using probabilistic arguments.