Let A and B be two random variables both independent from another random variable C. If A is independent from B, is A*B also independent from C? And if A and B are no independent from each other?
If all you have is pairwise independence then there is a counterexample. Suppose the following four cases each have probability $\frac14$:
A B C AB 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1
Then $A$ is independent of $C$ and $B$ is independent of $C$, and $A$ is independent of $B$.
But $AB$ and $C$ are not independent as $\mathbb P(AB=1\mid C=0)=\frac12\not= 0=\mathbb P(AB=1 \mid C=1)$
In this example $A$, $B$ and $C$ are pairwise independent as suggested by the question, but are not mutually independent. If they had been mutually independent then it would also follow that $AB$ would be independent of $C$. A slightly weaker condition is that if $A$ and $B$ were jointly independent of $C$ then it would follow that $AB$ would be independent of $C$ even if $A$ and $B$ were not independent of each other.