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:
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.