  An exact $r$-coloring of a set $S$ is a surjective function $c:S\to [r]$. The
rainbow number of a set $S$ for equation $eq$ is the smallest integer $r$ such
that every exact $r$-coloring of $S$ contains a rainbow solution to $eq$. In
this paper, the rainbow number of $\Z_p$, for $p$ prime and the equation
$a_1x_1 + a_2x_2 + a_3x_3 = b$ is determined. The rainbow number of $\Z_{n}$,
for a natural number $n$, is determined under certain conditions.
