Let $A$ and $B$ be arbitrary finite groups with $(|A|,|B|)=1$. Let $M(G)$ be the Schur multiplier of the group $G$. Problem 5A.8.(b) in Isaacs' Finite Group Theory asks us to show that $M(A \times B) \cong M(A) \times M(B) $.
We have that $|M(A \times B)| \ge |M(A)| |M(B)| $ by (a). I anticipate an argument using Schur-Zassenhaus, but I am having trouble applying the coprime condition to the structure of the extension.