Chinese remainder theorem for unit groups

For coprime $m,n$, reduction gives $(\mathbb{Z}/mn\mathbb{Z}{)}^{\times }\cong (\mathbb{Z}/m\mathbb{Z}{)}^{\times }\times (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$.

The order of a unit modulo a product of coprime moduli is the least common multiple of the orders of its components.