Problem (Algebra):
If $x, y$, and $z$ are positive integers such that $x y=20$ and $y z=12$, compute the smallest possible value of $x+z$.

Solution:
Note that $x$ and $z$ can each be minimized by making $y$ as large as possible, so set $y=$ $\operatorname{lcm}(12,20)=4$. Then $x=5, z=3$, and $x+z=\mathbf{8}$.

Final answer: 8