Intersection of Congruence Classes/Corollary

Corollary to Intersection of Congruence Classes Modulo m
Let $\mathcal R_m$ denote congruence modulo $m$ on the set of integers $\Z$.

If $m \perp n$ then $\mathcal R_m \cap \mathcal R_n = \mathcal R_{m n}$.

Proof
By Intersection of Congruence Classes Modulo m:


 * $\mathcal R_m \cap \mathcal R_n = \mathcal R_{\operatorname{lcm} \left\{{m, n}\right\}}$

$m \perp n$ means $\gcd \left\{{m, n}\right\} = 1$.

From Product of GCD and LCM it follows that $\operatorname{lcm} \left\{{m, n}\right\} = m n$.

Hence the result.