Isomorphism of $\mathbb{Z}_{30}$ to $ \mathbb{Z}_2\times \mathbb{Z}_3\times \mathbb{Z}_5$

Question

Hello I have a difficulty with this exercise the idea is to show $$\begin{align} \phi: \mathbb{Z}_{30} &\to \mathbb{Z}_2\times \mathbb{Z}_3\times \mathbb{Z}_5 \\ [x]_{30} &\mapsto ([x]_2,[x]_3,[x]_5) \end{align}$$ is surjective but what I was able to show is that $$\begin{align} \phi_1: \mathbb{Z}_{30} &\to \mathbb{Z}_2\times \mathbb{Z}_{15} \\ [x]_{30} & \mapsto ([x]_2,[x]_{15}) \end{align} $$ is surjective I wanted to consider $$\begin{align} \phi_2 : \mathbb{Z}_2\times \mathbb{Z}_{15} & \to \mathbb{Z}_2\times \mathbb{Z}_3\times \mathbb{Z}_5 \\ ([x]_2,[x]_{15}) & \mapsto ([x]_2,[x]_3,[x]_5) \end{align}$$ and show that is surjective but but I can't show $\phi_2$ is surjective I need help to tackle this exercise

Answer 1

Let $(b_1,b_2)\in \mathbb{Z}_{2}\times \mathbb{Z}_{15} $ and consider $ a=-14b_2+15b_1$ with have $ \phi(a)=(b_1,b_2)$ since $ [a]_2=b_1, [a]_{15}=b_2$