In mathematics, a covering system (also called a complete residue system) is a collection

${\displaystyle \{a_{1}(\mathrm {mod} \ {n_{1}}),\ \ldots ,\ a_{k}(\mathrm {mod} \ {n_{k}})\}}$

of finitely many residue classes ${\displaystyle a_{i}(\mathrm {mod} \ {n_{i}})=\{a_{i}+n_{i}x:\ x\in \mathbb {Z} \}}$ whose union contains every integer.