imc

Because the unpainted part of the floor covers half the area, then the painted rectangle covers half the area as well. Since the border width is 1 foot, the dimensions of the rectangle are $a-2$ by $b-2$. With this information we can make the equation: \begin{eqnarray} ab &=& 2\left((a-2)(b-2)\right) \\ ab &=& 2ab - 4a - 4b + 8 \\ ab - 4a - 4b + 8 &=& 0 \end{eqnarray} Applying Simon's Favorite Factoring Trick, we get \begin{eqnarray}ab - 4a - 4b + 16 &=& 8 \\ (a-4)(b-4) &=& 8 \end{eqnarray} Since $b>a$, then we have the possibilities $(a-4)=1$ and $(b-4)=8$, or $(a-4)=2$ and $(b-4)=4$. This allows for 2 possibilities: $(5,12)$ or $(6,8)$ which gives us $\boxed{2}$