Bézout's theorem states that a linear diophantine equation $ax+by=c$ has integer solutions if and only if $\gcd(a,b)|c$. Then is there a theorem regarding the existence of a unique integer solution for a system of two linear equations $ax+by=c$ and $px+qy=m$, where $a,b,c,p,q,m$ are positive integers?
Additionally, can the theorem be generalized to the condition where there are more than two unknowns?
For two equations the problem actually becomes very simple as the qeustion is equivalent to whether or not there exists a vector $\begin{pmatrix}x\\y\end{pmatrix}\in\mathbb{Z}^2$ with $$\begin{pmatrix}a&b\\p&q\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}c\\m\end{pmatrix}.$$
If the determinant of $A=\begin{pmatrix}a&b\\p&q\end{pmatrix}$ is non-zero, we have a unique real solution, which we can easily calculate via calculating $A^{-1}$. If this solution is already integer, we have a unique integer solution, else we dont.
If the determinant of $A$ is zero, there are two more cases to differentiate. Either $a=b=p=q=0$, in which case $m=c=0$ as well and there are infinitely many integer solutions.
Or the determinant of $A$ is zero and at least one of $a,b,p,q$ is non-zero. In that case the kernel of $A$ is $1$-dimensional and the equations therefor admit either no or infinitely many real solutions. If there are infinitely many real solutions and $s$ is such a solution, then the set of all real solutions can be described by $$\left\lbrace v\in\mathbb{R}^2\ \Big|\ v=s+w\text{, where }w\in\text{ker}(A)\right\rbrace.$$ Since the kernel is 1-dimensional, it is generated by a single vector $k\in\mathbb{R}^2$, thus in the last case the set of equations admits an integer solution if and only if there exists a $\lambda\in\mathbb{R}$ with $$s+\lambda \cdot k\in\mathbb{Z}^2,$$ and it is unique if and only if there is no $\mu\in\mathbb{R}$ with $\mu\cdot k\in\mathbb{Z}^2$. Both of those equations are easy to solve.
A necessary condition is: If we use Hermite normal form as mentioned in comments by @BillDubuque, we can assume $Ax = b$ with $A$ an upper triangular integer matrix.
A sufficient condition for the solution to exist: $$a_{ii} \ \ | \ \ gcd(a_{i,i+1},....,a_{i,n},b_i)$$.
A necessary condition for solution to exist: $$gcd(a_{ii},a_{i,i+1},....,a_{i,n}) \ \ | \ \ b_i $$.
If $a_{ii} = gcd(a_{i,i+1},....,a_{i,n})$ then the above necessary and sufficient condition become the same i.e., the solution exists iff $a_{ii} \ | \ b_i$ in this specific case. Need a necessary and sufficient condition in general though.
Good Luck !
