You probably got stuck because the statement looks like it might be nearly a tautology. It tempts you into hoping that it can be proven just by understanding the abstract properties of divisibility. For example, it's clearly true that $a$ and $b$ both divide $ab$. However, it turns out that that isn't enough!
As Brian has pointed out, you can call upon the least common multiple and greatest common divisor or $a$ and $b$. In fact, you almost have to do so! The existence of the lcm and gcd is a nontrivial property of the natural numbers $\mathbb N$. There are other number systems in which divisibility is meaningful but lcms and gcds don't always exist. For an example, see https://en.wikipedia.org/wiki/Greatest_common_divisor#The_gcd_in_commutative_rings.
So the statement isn't just about divisibility, and it's not just about partially ordered sets. It's an interesting property of $\mathbb N$. Therefore, no matter how clever one is, there is no hope of proving the statement without saying something about $\mathbb N$!
(For your purposes, it's probably enough to say "gcds and lcms exist in $\mathbb N$" without proof. I just wanted to point out why the problem is harder than it appears.)