Recently I've researched some positional notations. However, some of them I've invented can't express every natural number. I want to determine which of them can express every natural number, and which cannot, so that I can concentrate on those that can.
Let me define the positional notations I'm researching first. Consider a sequence of complete residue systems $A_1, A_2, \dots$ modulo $m_1, m_2, \dots$, respectively. (A complete residue system modulo $m$ is a set of size $m$, no two of which are congruent modulo $m$.) A number in this positional notation can be expressed in the form $$\overline{ a_n \dots a_2 a_1 } = a_n \prod_{ i=1 }^{ n-1 } m_i + \dots + a_2 m_1 + a_1$$ where for every $1 \le i \le n$, $a_i \in A_i$.
For example, decimal is $\{ 0, \dots, 9 \}, \dots$; balanced ternary, $\{ -1, 0, 1 \}, \dots$; negabinary, $\{ 0, 1 \}, \{ -1, 0 \} \dots$ (here ellipses denote repeating the whole sequence).
All the residue systems must contain $0$s, so that we can add leading zeros.
Some of the positional notations cannot express every natural number. The positional notation $\{ 0, 3 \} \dots$ cannot express $1$, for example. Are there any easy ways to determine if a positional notation can express every natural number?
I know that if each residue system contains consecutive integers including $0$ and $1$, then it can express every natural number, but I have no idea with universal cases.