Our goal in our Abstract Algebra class is to show that the ring $<\mathbb{Z}_{rs},+,\cdot>$ is isomorphic to the ring $<\mathbb{Z}_r\times \mathbb{Z_s},+,\cdot >$ where $gcd(r,s)=1$. We have no problem in showing that the map $\phi:\mathbb{Z}_{rs}\to \mathbb{Z}_r\times \mathbb{Z_s}$ defined by $\phi(m\cdot 1)=m(1,1)$ is (1) one-one and (2) satisfies the ring homomorphism property. But a problem arise when showing that the mapping defined is surjective or onto.
To show that $\phi$ is onto we need to show that for any $(m,n)\in \mathbb{Z}_r\times \mathbb{Z_s}$ there must be $x\in \mathbb{Z}_{rs}$ such that $\phi(x)=(m,n)$. But unfortunately we did not know how to find that $x$.
What is that $x$? Any help will be highly appreciated.