Product of gcd and lcm for multivariate polynomials

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.