Past exam of the short options course of the University of Oxford / 2021 / 1 / a / ii / Solution

Factor $Q={\prod }_i{p}_i$ and $Q^{\prime }={\prod }_j{q}_j$, retaining repeated prime factors. If $P$ has a common prime factor with $Q{Q}^{\prime }$, it has one with at least one of $Q,Q^{\prime }$, and both sides of the desired equality are zero. Otherwise the definition of the Jacobi symbol givesAn empty product is $1$, so the argument includes either denominator being $1$.