  In this work, we investigate the fewest number of colors needed to guarantee
a rainbow solution to the equation $x_1 + x_2 = k x_3$ in $\mathbb{Z}_n$. This
value is called the Rainbow number and is denoted by $rb(\mathbb{Z}_n, k)$ for
positive integer values of $n$ and $k$. We find that $rb(\mathbb{Z}_p, 1) = 4$
for all primes greater than $3$ and that $rb(\mathbb{Z}_n, 1)$ can be
deterimined from the prime factorization of $n$. Furthermore, when $k$ is
prime, $rb(\mathbb{Z}_n, k)$ can be determined from the prime factorization of
$n$.
