Intersection of Congruence Classes/Corollary

Corollary to Intersection of Congruence Classes
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:


 * $\mathcal R_m \cap \mathcal R_n = \mathcal R_{\lcm \set {m, n} }$

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

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

Hence the result.