A family of residue classes $a_i (\mod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\mod n_i)$. The known examples are:
$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 5 (\mod 6),\ 7 (\mod 12)$
$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 3 (\mod 8),\ 7 (\mod 12),\ 23 (\mod 24)$
The proof of that the above families are each a covering system of integers are not difficult.
My question is the other side, i.e., that how can we construct a covering for integers from the given numbers for example $2,3$ with their multipliers as moduli?