Multiplicative Question

**Janu42** · October 18th 2010, 10:06 PM

Assume $\sigma$ is multiplicative

a) Let p be a prime and a $\in Z^+$ . Prove that for an odd prime p, $\sigma(p^a)$ is odd if, and only f, a is even. Also prove that for p = 2, $\sigma(2^a)$ is odd for all a $\in Z^+.$

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