Given $m_0,m_1,\cdots, m_{r-1} \in R$, here $R$ is a euclidean domain. we want to find $f$ which satisfies $f \cong v_1 $mod $m_0$, $f \cong v_2 $mod $m_1$ and so on $f \cong v_{r} $mod $m_{r-1}$.
My Attempt :let us rewrite these equations as , $f = v_1 + m_0t_1$,$f = v_2 + m_1t_2$ and so on $f = v_{r} + m_{r-1}t_{r}$. So now I have system of equations , now I can use gaussian elimination method to find $f$. Does it always give a valid solution?
Edit : Let $R = \mathbb{Z}$, in this case while solving the system of equation I may get a problem because in this process I may need to multiply a row with $\frac{a}{b} \in \mathbb{Q}$ not in $\mathbb{Z}$. So I think ( I am not sure) we can't use gaussian elimination here.
You are right, we can encouter problems even for $\Bbb Z$. For instance, let $r=3$, $m_0=2$, $m_1=3$, $m_2=5$, $v_1=1$, $v_2=2$, and $v_3=3$. Then we have equations $f=1+2t_1$, $f=2+3t_2$, and $f=3+5t_3$. Eliminating $f$ we obtain equations $2t_1=1+3t_2$ and $3t_2=1+5t_3$. Eliminating $3t_2$ we obtain a equation $2t_1-1=1+5t_3$. Pick its solution, $t_1=1$, $t_3=0$. But then there exists no integer $t_2$ such that $2=2t_1=1+3t_2$.