Proving that an arithmetic function is multiplicative, but not completely multiplicative? 
Define an arithmetic function $\rho$ by $\rho(1)=1$ and $\rho(n)=2^m$ where m is the number of distinct prime numbers in the prime factorization of n. Prove that $\rho$ is multiplicative, but not completely multiplicative. 

This is my first introduction to arithmetic functions so I'm not quite sure how to prove these kinds of things. I can plug numbers into the $\rho$ function,  but how can I show this through a formal proof?

Just show that $\rho(2)^2\ne \rho(2^2)$. Clearly $\rho(2)=\rho(4)=2$, so you're done.
