Connection between rank and matrix product I have a problem understanding the following:

Let $A$ be an $m \times n$ matrix and let t $\in \mathbb{N}$. Prove that
  $\operatorname{rank}(A)\leq t$ if and only if there exists an $m \times t$ matrix $B$ and a $t \times n$ matrix $C$ so that $A = BC$.

I know what a rank is but I can't make a connection between the rank and the existence of two matrices such that $A = BC$.

There is a Theorem that states that 
Rank(XY)<=min(rankX, rankY)
So 
Rank(BC)<=min(rankB, rankC)
So if t is less than n and m. Then it could be the rank of either B or C 
Rank(BC)<=t. 
I hope this helps. 
