Lemma: We can build any $2\times n$ piece for any $n\geq5$ or $n=3$. The example is trivial for odd $n$. For $n$ even, just put a $2\times3$ and a $2\times(n-3)$ side by side. Assume the board has $n$ rows and $m$ columns. Now, to solve the problem, separate in two cases: $n$ even. Then, the construction is to place $\frac{n}{2}$ pieces $2\times m$. $n$ and $m$ both odd. Clearly, by adding pieces $2\times k$, this is equivalent to solving ANY small board of odd sides. One can find an example for $5\times11$.