When linear equation system $Ax = b$ has integer solutions?

Question

$A$ is a rational matrix, and $b$ is a rational vector. Under which condition the system will have integer solutions?

Answer 1

I am sure my answer is not even a bit near perfect, but I still want to give a headstart for everyone reading this question.

First of all, we can multiply all entries of $A$ and $b$ by an integer without affecting integral solutions, since $\sum a_ix_i=b_i$ is equivalent to $\sum Na_ix_i=Nb_i$.

Also, notice that $gcd(a_{i1},a_{i2},\dots,a_{in})|b_i$, or else with Euclidean Algorithm, there will not be any solution. With this, we can state that for $1$ row, this question is easy (with Enclidean Algorithm above), but with multiple rows I still have no idea on how to effectively tackle it.