$\frac{S}{d_{250}}\ is\ the\ 250th\ divisor\ in\ increasing\ order,\ so\ he \ has\ maximum\ 250\ divisors.\ We\ deduce\ that\ d_{250}\ has\ minimum\ 4000\ divisors.\\ An\ example\ for\ this\ is\ S=p^{3999}\cdot 2^{249},\ where\ p>2^{249}\ is\ prime\ number.\ Then\ d_{250}=p^{3999},\ and\ it\ has\ 4000\ divisors. $