Deductions from Carmichael's theorem 1.) May I seek your advice on how to prove the following assertion(without recourse to (2)): If $ 2014 \equiv 14 \ (\text{mod} \ 2000), $ then $2014^{2014} \equiv 14^{14} \ (\text{mod 2000}).$ 2.) Also, how we prove from Carmichael's theorem that, "If $a \equiv b \ (\text{mod m})$ and $c \equiv d \ (\text{mod} \ \lambda(m) ),$ then $a^c \equiv b^d \ (\text{mod m})$ ", where $\lambda (m)$ is defined as such: < Thank you.

$\lambda(n)$ is the smallest number such that for any k we have $k^{\lambda(n)}\equiv 1 \bmod n$ for any $k$. suppose we have $b\equiv c \bmod \lambda n$. Then $b=c+x\lambda(n)$ for some integer x. Then we have $k^{c}=k^{b+x\lambda(n)}=k^b*(k^{\lambda(n)})^x\equiv k^b*1^x\equiv k^b \bmod n$