BALKAN 2001 Problem 1

Let $a,b,n$ be positive integers such that $2^n - 1 =ab$ . Let $k \in \mathbb N$ such that $ab+a-b-1 \equiv 0 \pmod {2^k}$ and $ab+a-b-1 \neq 0 \pmod {2^{k+1}}$ . Prove that $k$ is even.