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

Let be an matrix and suppose that is an matrix such that every column of is in the column space of .

Prove that there is a matrix such that ( would be an size matrix).

[HINT: we know that the equation has a solution if and only if .]

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