The solution set of $a \equiv b \pmod m$ is given by the set of values satisfying $b \pmod m$.
The reason is that the stated congruence is actually an equality given by : $\exists k,l \in \mathbb{Z}$, s.t. $a - km = b - lm$.
An easy example is : $17 \equiv 2 \pmod 5$, with solutions being given by the infinite set of values of $k,l$:
1. $(k=3, l =0)$ => $17 - 3\cdot 5 => 2$, $2 - 0 => 2$
2. $(k=0, l =-3)$ => $17 - 0 => 17$, $2 +15 => 17$
3. $(k=1, l =-2)$ => $17 - 5 => 12$, $2 + 10 => 12$
4. $(k=-1, l =-4)$ => $17 +5 => 22$, $2 +20=> 22$
I am trying to find relation between the different values of $k,l$ once one particular solution is arrived by the trivial means of having $k=0$, or $l=0$.
In the above example, it seems that the different values of $k,l$ satisfy the relation: $k - l = 3$.
My question : is there any relation among the values of $k,l$, which is a general one.
$a-km=b-lm\\ a-b=km-lm\\ a-b=m(k-l)\\ k-l=\frac{a-b}{m}\text.$