Past exam of the options II course of the University of Oxford / 2013 / C2 / b

Factor $a={\prod }_i{q}_i$ and $b={\prod }_j{p}_j$, listing prime factors with repetitions. Coprimality ensures $q_i\ne p_j$. By the definition of the Jacobi symbol and multiplicativity of the Legendre symbol,For odd integers $r,s$,since the difference is $(r-1)(s-1)/2$, an even integer. Inductively the two sums in the exponent are congruent to $(a-1)/2$ and $(b-1)/2$ modulo two. HenceIf either number is one, the empty product convention proves the same identity. This extends the prime law to arbitrary coprime positive odd denominators.