About a new type of congruence system.

Question

If $A\equiv B\pmod m$ then m|(A-B). What if you used a 2 coordinate congruence where $C\equiv D\pmod {m,n}$ then there exists R,S such that (C-D) = m R+n S, where R and S are coprime and |mR+nS| > gcd(m,n). So if $A\equiv B\pmod {m,n}$ then $A^2\equiv B^2\pmod {m,n}$. Is this a useful extension of congruences? (If $A\equiv B\pmod {m,n}$ then $Ar\equiv Br\pmod {m,n}$)

Answer 1

The set of integers of the form $\{ Rm + Sn : R,S \in \mathbb{Z} \}$ consists of all integer multiples of $g=\mathrm{gcd}(m,n)$. Indeed, since $g$ divides both $m$ and $n$, it divides any integer of the form $Rm+Sn$. The converse follows from the well-known fact that $g = Rm+Sn$ for some integers $R,S$. You can see proofs on this Wikipedia page.

As a conclusion, we obtain that $a \equiv b \pmod{m,n}$ iff $a \equiv b \pmod{\mathrm{gcd}(m,n)}$.