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Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample.
a) If $\operatorname{det}(A)=\pm 1$, then for any vector $y$ with integer elements there is a vector $x$ with integer elements that solves $A x=y$.
b) If $\operatorname{det}(A)=2$, then for any vector $y$ with even integer elements there is a vector $x$ with integer elements that solves $A x=y$.

c) If all of the elements of $A$ are positive integers and $\operatorname{det}(A)=+1$, then given any vector $y$ with non-negative integer elements there is a vector $x$ with non-negative integer elements that solves $A x=y$.
d) If the elements of $A$ are rational numbers and $\operatorname{det}(A) \neq 0$, then for any vector $y$ with rational elements there is a vector $x$ with rational elements that solves $A x=y$.
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