Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / C / C2 / b / ii / Solution

If $b,c$ have a common prime factor, both Jacobi symbols are zero, and the identity holds. Otherwise write $b={\prod }_i{p}_i$, $c={\prod }_j{q}_j$ with multiplicity. All cross-pairs of primes are distinct. Multiplying quadratic reciprocity over these pairs givesThe exponent factors as a product of two sums. The parity identity from part (i) makes it congruent modulo two to $[(b-1)/2][(c-1)/2]$. ThereforeThe same formula includes a denominator equal to one by the empty-product convention.