Gauss lemma for arbitrary commutative ring

Question

Part (iv) of exercise #2 for chapter 1 in Atiyah and Macdonald's book Introduction to Commutative Algebra asserts that if $f, g \in A[x]$ are primitive then $fg$ is primitive. We know that this is true when the coefficient ring $A$ is a UFD in which case it is the statement of Gauss' lemma. I have tried hard to show it for arbitrary ring as the problem says but did not manage to do that. So my question is: Though I do not doubt the author, is Gauss lemma really valid for arbitrary commutative rings with unity and can I have some hint?

Answer 1

Hint. Assume the contrary and then reduce all data modulo a maximal ideal.