I want to know if a given system of linear integer equations has an integer solution. I know it is the case if and only if it has a solution modulo $n$ for all integer $n$. What I do not know is when such a system admits a solution modulo a generic $n$.
For example, a single linear equation $\sum a_i X_i = b$ has got a solution modulo $n$ if and only if the greatest common divisor of its coefficients $a_i$ and n divides $b$. Can something similar be said about a system? I'd be satisfied with any answer. For instance just some link or refference to look up would be enough. Thanks.