Prove there is a matrix B

Let $\displaystyle A$ be an $\displaystyle m\times n$ matrix and suppose that $\displaystyle X$ is an $\displaystyle n\times p$ matrix such that every column of $\displaystyle X$ is in the column space of $\displaystyle A$. Prove there is a matrix $\displaystyle B$ such that $\displaystyle AX=B$

[Hint: we know that the equation $\displaystyle Ax=v$ has a solution if and only if $\displaystyle v\in col(A)$]

I don't know if this is exactly how the question is meant to be said. Our Prof. made a severe typo in the original, so I'm asking him to simply rewrite the question in its intended form rather than just provide us with a quick correction (which could be easily misinterpreted).