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In this work, we investigate the fewest number of colors needed to guaranteea rainbow solution to the equation $x_1 + x_2 = k x_3$ in $\mathbb{Z}_n$. Thisvalue is called the Rainbow number and is denoted by $rb(\mathbb{Z}_n, k)$ forpositive 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 bedeterimined from the prime factorization of $n$. Furthermore, when $k$ isprime, $rb(\mathbb{Z}_n, k)$ can be determined from the prime factorization of$n$.

Rainbow numbers for $x_1+x_2=kx_3$ in $\mathbb{Z}_n$