I came across this question in my course.
Consider a system AX=B of n linear equations in n unknowns where A and B have integer entries. Prove or disprove : If the system has an integer solution, then it has a solution in F(p) for all p.
Well, I could see that I have to disprove it. Since the determinant could be 0 (mod p). In which case the system needn't have a solution. But how exactly do i present a formal proof for this?
One more question,
It says in my book for the system AX = B
( 8 3 = A
and B = (3 -1)t
(I hope you understand the matrix notations.) , there is a solution in F(7), even though det(A)=42 ~0 (mod 7). How is this ?