Prove that for all positive integers $n \geq 17$ such that $\left\lfloor \frac{n}{4} \right\rfloor \equiv 0 \pmod 2$, there exists positive integers $a, b$ and an even positive integers $c$ such that the following are satisfied:
$$ a+b = 4c-1$$ and $$\left\lfloor \frac{n}{4}\right\rfloor + c \equiv 0 \pmod {ab}$$
To me, this seems logical but I've spent the past month trying to prove it but haven't been able to make any progress. I feel really stumped. Any help is appreciated, thanks!
For $n$ as in the problem statement let $c:=\lfloor \frac n4\rfloor$, $a=2c$, $b=2c-1$.