It's given that if a positive integer $n$ is Not a power of two, then $n$ must have an odd prime factor, meaning $$n = pr, p>2, 1\leq r< n $$ Is it really this trivial? There's a proof that uses this result, without even giving an explanation why it's true. If we assume $p=2$ why is that a contradiction?
If $r=1$, then $n = 2\cdot 1$, which is a power of two, hence contradiction.
Assume $n = 2k$ is a power of two, then $n=2(k+1) = 2k+2\cdot 1$, but there's no contradiction.