Product of gcd and lcm for multivariate polynomials

Question

This maybe trivial but I don't know how to conclude the proof. Consider the ring of multivariate polynomials with field coefficients $K[X_1,\dots,X_n]$. Take two nonzero polynomials $F$ and $G$ and prove:

$$FG=\gcd(F,G)lcm(F,G)$$

Since such ring is not Bezout I don't know how to prove this. I managed to prove that their gcd and their lcm both divide their product but I don't know how to do the converse relation.

Answer 1

The ring is a UFD so a GCD domain. As proved here the identity holds true in any GCD domain.

It can also be proved via prime factorizations, but that is less general than said proof using gcd laws.