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Let $n$ be a positive integer and $\mathbf{A}$ an $n$-by-$n$ matrix with integer entries. Suppose that $k:=\big|\det(\mathbf{A})\big|$ is nonzero. How many $n$-by-$1$ column vectors $\mathbf{x}\in\mathbb{Q}^n$ each of whose entries lies within the interval $[0,1)$ are there with the property that $\mathbf{A}\,\mathbf{x}=\mathbf{b}$ for some $n$-by-$1$ column vector $\mathbf{b}\in\mathbb{Z}^n$?

When $n=1$, the answer is trivially $k$. According to this thread, when $n=2$, the answer is also $k$. I believe that, in general, the answer is $k$. Geometrically, the image of $\mathbf{x}\mapsto \mathbf{A}\,\mathbf{x}$ for $\mathbf{x}\in[0,1)^n$ is an $n$-dimensional hyper-parallelipiped $H\subseteq \mathbb{R}^n$ with integral vertices and with volume $k$. However, I am not certain how to calculate the number of integral points inside and on the boundary of $H$. I remember that there may be a theorem relating the volume and the number of integral points within and on the boundary of a convex polytope whose vertices are integral points. What I do not remember is the name of the theorem and what it states exactly.

I found out what I was referring to, and it was the Ehrhart Polynomial Theorem. The answer is indeed $k$. However, I will still greatly appreciate a number-theoretic or algebraic solution.

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You are basically asking for the number of (standard) lattice points inside the fundamental parallelepiped $F$ of the lattice generated by the matrix $A$. Since the vertices of $F$ are on the standard lattice, the number of lattice points in $F$ is the same as the number of points in $F+x$ for any translate of $F$ by an integer vector $x$. Therefore, if you multiply (scale) $F$ by an integer $r$, then the number of points in $rF$ is $r^n$ times the number of points in $F$.

On the other hand, for any convex body $K\subset \mathbb{R}^n$, $$ \lim_{r\to\infty} \frac{\#\{ rK \cap \mathbb{Z}^n \}}{r^n} = \rm{Vol}(K).$$

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