structural:summarization, stylistic:tone
Lattice-theoretic

The positive integers may be partially ordered by divisibility: if a divides b (that is, if b is an integer multiple of a) write a ≤ b (or equivalently, b ≥ a). (Note that the usual magnitude-based definition of ≤ is not used here.)

Under this ordering, the positive integers become a lattice, with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them:

The following pairs of dual formulas are special cases of general lattice-theoretic identities.

It can also be shown that this lattice is distributive; that is, lcm distributes over gcd and gcd distributes over lcm:

This identity is self-dual: