Let $m$ and $n$ be relatively prime positive integers. Define $\alpha : \mathbb{Z}_{mn} \rightarrow \mathbb{Z}_m \times \mathbb{Z}_n$ by $\alpha([a]_{mn}) = ([a]_m,[a]_n)$.
Prove that $\alpha$ is injective.
My attempt:
To prove $\alpha$ is injective, $(\forall a_1, a_2 \in \mathbb{Z})(\alpha([a_1]_{mn})=\alpha([a_2]_{mn}) \rightarrow [a_1]_{mn} = [a_2]_{mn})$ needs to be shown.
Therefore, $\alpha([a_1]_{mn})=\alpha([a_2]_{mn}) \implies ([a_1]_m, [a_1]_n) = ([a_2]_m, [a_2]_n)$, so $[a_1]_m = [a_2]_m$ and $[a_1]_n = [a_2]_n$. So, $a_1 = a_2 + c_1m$ and $a_1 = a_2 + c_2n$.
However, I don't know how to show that $a_1 = a_2 + c_3mn \implies [a_1]_{mn} = [a_2]_{mn}$.
Am I taking the correct approach at this? What am I overlooking?