Why is RSA encryption exponent always odd/never even?

Question

I remember my professor mentioning that RSA encryption fails when $e$ is odd, but cannot seem to figure out why it is so, and can't find a proof in a textbook/online. I tried verifying it by proof-by-contradiction by considering the even case, i.e. $a^{2t}\pmod n$ (where n=pq), but don't see where the contradiction would be coming from. Do I need to bring in Euler's totient $\varphi(n)=\varphi(pq)=\varphi(p-1)(q-1)$ somehow?