it's actually true for an obvious reason: let write AX = B as a system of equations: for any integer let be

then we'll have thus is a solution modulo p.

does not imply that the equation has no solution. so there's nothing strange about your example.

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 ?