Define the notion of a Fermat, Euler, and strong pseudo-prime to the base $b$, where $b$ is an integer greater than $1 .$

Let $N$ be an odd integer greater than 1. Prove that:

(a) If $N$ is a prime number, then $N$ is a strong pseudo-prime for every base $b$ with $(b, N)=1$.

(b) If there exists a base $b_{1}$ with $1<b_{1}<N$ and $\left(b_{1}, N\right)=1$ for which $N$ is not a pseudo-prime, then in fact $N$ is not a pseudo-prime for at least half of all bases $b$ with $1<b<N$ and $(b, N)=1$.

Prove that 341 is a Fermat pseudo-prime, but not an Euler pseudo-prime, to the base $2 .$