Solving Systems of Congruences when mods not pairwise relatively prime

CantorSet

Hi folks,

The CRT says there's a unique solution to the system of congruences

[itex] x = a [/itex] (mod m)
[itex] x = b [/itex] (mod n)
[itex] x = c [/itex] (mod p)

in (mod mnp) when [itex] m, n, p [/itex] are pairwise relatively prime. But what if [itex] m, n, p [/itex] are NOT pairwise relatively prime. Is there a systematic way to solve these cases?

Eynstone

The system may not have a solution if the moduli are not pairwise coprime.We can, of course,solve two equations at a time modulo the lcm & try to patch up the solutions... I don't know how to answer this best.