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

$x = a$ (mod m)
$x = b$ (mod n)
$x = c$ (mod p)

in (mod mnp) when $m, n, p$ are pairwise relatively prime. But what if $m, n, p$ 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.