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

where

( 8 3 = A

2 6)

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 ?