Primary decompositions and divisorial fractional ideals

Question

Consider a divisorial (fractional) ideal $\mathfrak{a}$ over some normal Noetherian domain $A$ with fraction field $K$. Then it can be shown that $\mathfrak{a}$ is a product of integer powers of primes of codimension one. Part of the proof given in my algebraic geometry class uses the fact that if $b$ is a nonzero element of $A$ with $b\mathfrak{a}\subset A$, then the associated primes of $b\mathfrak{a}$ have height one, and moreover $b\mathfrak{a}$ is the product of powers of these primes. I'm having trouble seeing why this is the case, though; my professor just said the first follows from the definition of a divisorial ideal and the second from normality of $A$. Any help or references would be appreciated. (I should mention that I was able to find a different proof in Eisenbud's commutative algebra text, but that proof only applies in the case that $A$ is locally factorial and $\mathfrak{a}$ is invertible.)