Let $Ax=B$ be a system of linear diophantine equations, where $A$ is a full rank $n \times 2n$-matrix with integer entries. In the case $n=1$ we have solutions parameterized by $\mathbb{Z}$ iff $gcd(a\_{11},a\_{12})$ divides $b\_{11}$. Is there a similar statement for arbitrary $n$ of the form "We get solutions paramterized by $\mathbb{Z}^n$ iff *condition on $B$*"


EDIT: Since I know basically nothing about diophantine equations, it could be that this question is far from being research level.. if so, just tell me and I will delete it and bring it up in math SE



Solvability conditions for linear system of diophantine equations
Surely, the Smith normal form does it all. But if you need a more concrete condition, here is one. 


Let $X$ be the set of all $n\times n$ minors of $A$, and let $Y$ be the set of all $n\times n$ minors of $(A\,| B)$. Then the equivalent condition is that $\gcd(X)=\gcd(Y)$. Indeed, this condition is equivalent if the system is in the Smith form, and moreover it is preserved by $SL(n,\mathbb Z)$-transforms.


