(a) State and prove the Chinese remainder theorem.

(b) Let $N$ be an odd positive composite integer, and $b$ a positive integer with $(b, N)=1$. What does it mean to say that $N$ is a Fermat pseudoprime to base b? Show that 35 is a Fermat pseudoprime to base $b$ if and only if $b$ is congruent to one of $1,6,29$ or $34(\bmod 35)$.