A form of Chinese remainder theorem

Question

How can we solve equations of the form $c \equiv a \mod b$ for finding the c?

Also, sometimes $c$ can be two different numbers, one negative and one positive, when is that possible and how does it work?

Answer 1

This has nothing to do with the Chinese remainder theorem.

This is just one of the basic definitions in modular arithmetic.

$c \equiv a \mod b$ if and only if $b \mid c-a$.

The Chinese remainder theorem help you solve two congruences in relatively prime moduli simultaneously. This isn't the case here. Moreover, $c \equiv a \mod b$ isn't an equation in modular arithmetic, because you're not looking for any number that when it's plugged in it satisfies the equation.