Prove there is a matrix X

I got the same question, but there was a typo in it. I think it should read like this:

Let $A$ be an $m\times n$ matrix and suppose that $X$ is an $n\times p$ matrix such that every column of $X$ is in the column space of $A$ .
Prove that there is a matrix $B$ such that $AX=B$ ( $B$ would be an $m\times p$ size matrix).
[HINT: we know that the equation $Ax=v$ has a solution if and only if $v\in col(A)$ .]

An addition to the hint was provided that for any column (i.e. $v$ ) of $B$ , there is always a solution, $x$ , to $Ax=v$ , so we are to apply this successively to each column and determine how many $x$ 's we obtain.