Show that $(\Bbb{N}, |)$ is a distributive lattice.

Question

Show that the set of Natural numbers with divisibility form a distributive Lattice where for any $x, y\in\mathbb{N}$ we have $x\wedge y = \operatorname{gcd}(x,y)$ and $x\vee y=\operatorname{lcm}(x,y)$?

I totally have no clue about this question..

Answer 1

To show distributivity its enough to show always $$a\vee t=b\vee t, a\wedge t=b\wedge t\to a=b$$ here we have $$(a\vee t)(a\wedge t) = at$$