By Cramer's rule, we know that element $x_i$ of solution matrix $X$ can found by the following formula:
$x_i=\frac{det(A_i)}{det(A)}$
assuming $det(A)=1$ , we get $x_i=det(A_i)$
(recall when applying cramer's rule, $A_i$ is a matrix formed by replacing the $i^{th}$ column in matrix $A$ with column matrix $B$)
It is already given that elements of $B$ are integral and all coefficients in equations are integral therefore elements of matrix $A$ are integral as well. As $A_i$ is made from elements of $A$ and $B$, therefore it's elements are integral as well.
Determinant of a matrix with integral elements will always be integral (you can prove this yourself) therefore $det(A_i)$ will also be integral. As $x_i= det(A_i)$ therefore $x_i$ is integral as well.
Hence we have shown that all elements of solution matrix X will be integral when $det(A)=1$.
So answer is (b)