Given two systems of congruences with one common parameter $A$ I need to find all values of $A$ for which both systems have the same solution sets. The systems are: $$x\equiv5\bmod6\\x\equiv A\bmod35$$and$$x\equiv A\bmod10\\x\equiv 14\bmod21$$
I solved it by first finding x in both systems, by CRT and then since the answers were both $\bmod 210$ I equated both solutions with that modulus and solved for $A$.
From the first system I got $x\equiv36A+35\bmod210$, and from the second one $x\equiv21A-70\bmod210$. Equating the $x$'s I got $A\equiv3\bmod6$. Is it correct? And what would I have to do if the moduli weren't equal? Would finding the LCA and multiplying both solutions be correct?