That's the application of the CRT:
and therefore
The isomorphism existence (defined by ) comes from the CRT. (where stands for )
So iff i.e. iff
And same thing with
Find all solutions, if any, to the system of congruences
x 7 (mod 9)
x 4 (mod 12)
x 16 (mod 21)
Solution manual states: We cannot apply the CRT directly, since the moduli are not pairwise relatively prime. (Got that on my own.) However, we can, using the CRT, translate these congruences into a set of congruences that together are equivalent to the given congruence. Since, we want x 4 (mod 12), we must have x 4 1 (mod 3) and x 4 0 (mod 4).
First question, how do we convert congruences? I can see that dividing the 4 and 12 by 4 could give 1 (mod 3), is that the correct procedure? But I don't see how that equates to 0 (mod 4), could you help me there?
Similarly, from the third congruence we must have x 1 (mod 3) and x 2 (mod 7). Why?
From here I think I can solve the following system with the excellent help I've received previously.
x 7 (mod 9)
x 0 (mod 4)
x 2 (mod 7).
Thanks for any explanations.