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

For a prime number $p$, multiplicativity of the Legendre symbol says $(P{P}^{\prime }/p)=(P/p)(P^{\prime }/p)$, including the case of a zero factor. Therefore, writing $Q={\prod }_i{p}_i$ with repetitions,This product definition still agrees with the stipulated zero when the numerator and denominator are not coprime. For $Q=1$ all three symbols are $1$.