let d,k be integers, with k even.
suppose d|2k
suppose d does not divide 2
suppose d does not divide k
show that d equals 2k.
(I'm really just trying to understand the 2nd last line, in this answer to the question: Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$)