Assume$\displaystyle \sigma$ is multiplicative

a) Letpbe a prime anda$\displaystyle \in Z^+$. Prove that for an odd primep, $\displaystyle \sigma(p^a)$ is odd if, and only f,ais even. Also prove that forp= 2, $\displaystyle \sigma(2^a)$ is odd for alla$\displaystyle \in Z^+.$

b) Let n $\displaystyle \in Z^+$. Prove that $\displaystyle \sigma(n)$ is odd if, and only if, eithernis a perfect square or twice a perfect square (i.e. $\displaystyle n \in {(m^2 \mid m \in Z^+)} \cup {(2m^2 \mid m \in Z})$