TUYMAADA 2017 S6

Let $\sigma(n)$ denote the sum of positive divisors of a number $n$ . A positive integer $N=2^r b$ is given, where $r$ and $b$ are positive integers and $b$ is odd. It is known that $\sigma(N)=2N-1$ . Prove that $b$ and $\sigma(b)$ are coprime.

(J. Antalan, J. Dris)