Past exam of the short options course of the University of Oxford / 2021 / 1 / b / i / Solution

The supplementary law for the Legendre symbol is $(-1/p)=(-1{)}^{(p-1)/2}$ for odd prime numbers $p$. For odd integers $a,b$,and the last term is even. Induction over the prime factors of $Q$ therefore gives ${\sum }_i(p_i-1)/2\equiv (Q-1)/2(\mathrm{mod}2)$. ConsequentlyFor $Q=1$ the exponent is zero.