Prove that a composite number $n$ is a Carmichael number if and only if $b^{n-1} \equiv 1 \mod n$ for all integers b with $(b,n)=1$
Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.