TITLE: $Ax=b$ has a solution over $\Bbb{F}_p$ for every prime $p~\implies$ existence of real solution??
QUESTION [8 upvotes]: $A$ be an $m\times n$ matrix and $b$ a $m\times1$ vector, both with integer entries. If $Ax=b$ has a solution over $\Bbb{F}_p$ for every prime $p$, is a real solution guaranteed?
I couldn't think of any easy way to begin this problem. Any hint please.
Bonus question: A matrix with all diagonal elements odd integers and all non diagonal elements even integers, is the identity matrix in $\Bbb{F}_2$. Does that imply it is invertible as a real matrix?

REPLY [3 votes]: For the bonus question, the condition implies the determinant is odd, hence not zero, hence the matrix is invertible over the reals (indeed, over the rationals).