An exact $r$-coloring of a set $S$ is a surjective function $c:S\to \{1,2,\cdots, r\}$. Given an equation $eq$, a solution in $S$ is a rainbow solution if each element is colored distinctly by the coloring $c$. 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 $eq$. The rainbow numbers of $\mathbb{Z}_p$, for prime $p$, for the equation $x_1 + x_2 = 4x_3$ are known to be either $3$ or $4$. This paper investigates which primes yield rainbow number $3$ or $4$. Additionally, the rainbow numbers of $\mathbb{Z}_n$ for this equation are discussed.