jmc

Instead of representing the quotient as $1\Delta$, we will represent it as $10+\Delta$. Our equation becomes $144/\Delta =10+\Delta$. Multiplying both sides by $\Delta$, we have a quadratic: $144=10\cdot \Delta +{\Delta }^2$. Rearranging the terms, we have ${\Delta }^2+10\cdot \Delta -144=0$. We can now factor this quadratic as $(\Delta +18)(\Delta -8)=0$ or use the quadratic formula: $\Delta =\frac{-10\pm \sqrt{10^2-4(1)(-144)}}{2}=-18\text{ or }8$.

Since $\Delta$ is a digit, $\Delta \ne -18$. Therefore, $\Delta =\boxed{8}$.