jmc

So, let there be $x$ pairs of $\text{textdollar}1$ socks, $y$ pairs of $\text{textdollar}3$ socks, and $z$ pairs of $\text{textdollar}4$ socks. We have $x+y+z=12$, $x+3y+4z=24$, and $x,y,z\ge 1$. Now, we subtract to find $2y+3z=12$, and $y,z\ge 1$. It follows that $2y$ is a multiple of $3$ and $3z$ is a multiple of $3$. Since sum of 2 multiples of 3 = multiple of 3, so we must have $2y=6$. Therefore, $y=3$, and it follows that $z=2$. Now, $x=12-y-z=12-3-2=\boxed{7}$, as desired.