jmc

Let $a$ be the desired number. We know that $$\begin{array}{rl}a& \equiv 0(\mathrm{mod}2)\\ a& \equiv 1(\mathrm{mod}3)\\ a& \equiv 2(\mathrm{mod}4)\end{array}$$ Note that $a\equiv 2(\mathrm{mod}4)$ automatically implies $a\equiv 0(\mathrm{mod}2)$, so only $a\equiv 1(\mathrm{mod}3)$ and $a\equiv 2(\mathrm{mod}4)$ need to be considered. The first few positive solutions of $a\equiv 2(\mathrm{mod}4)$ are $2,6,10$. While the first two do not satisfy $a\equiv 1(\mathrm{mod}3)$, luckily $\boxed{10}$ does!