Let $A$ be a ring of dimension $1$ and let $J$ be an ideal of $A$ that can be factored as a product of maximal ideals $J = P_1^{a_1} \dotsm P_s^{a_s}$. Let $M$ be any maximal ideal of $A$.
Then $J_M = (MA_M)^{a_i}$ if $M= P_i$ for some $i$, and $J_M = A_M$ if $M \neq P_i$ for any $i = 1,...,s$.
The second part is easy enough - get an element in $J\setminus M$ and it's invertible in $A_M$ so you get $J_M = A_M$.
The first part I am not so sure what to do. I can write down parts of the LHS and RHS as $J_M = J_{P_i} = S^{-1}J$ for $S = A \setminus P_i$, and $(MA_M)^{a_i} = (P_i A_{P_i})^{a_i} = (S^{-1} P_i)^{a_i}$ for the same $S$.
I imagine the answer will use the map $j^{-1} : Spec(S^{-1} A) \to \{ P : P \in Spec(A), P \subset A \setminus S \} $, or something like this, but I am not sure how to proceed.