Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / C / C2 / iv

For $a=2$ the displayed multiples are $2,4,\dots ,p-1$, already least positive residue classes. Those greater than $p/2$ correspond to $j>p/4$. Thus their number is $m=(p-1)/2-\lfloor p/4\rfloor$. For $p\equiv 1,3,5,7(\mathrm{mod}8)$, its parity is respectively even, odd, odd, even. Gauss's lemma therefore gives