Proving that a function is multiplicative. Let $f(x)$ be a polynomial with integral coefficients, and let $\psi(n)$ denote the numbers of values $f(0), f(1), ..., f(n-1)$ which are coprime to $n$. 
I must show that $\psi$ is multiplicative, meaning that:
$$\psi(mn) = \psi(m) \cdot \psi(n)$$
assuming $\gcd(m,n)=1$.
Furthermore, I must show that 
$$\psi(p^\alpha) = p^{\alpha-1} \cdot (p-b_p)$$
where $b_p$ is the number of integers $f(0), f(1), ..., f(p-1)$ which are divisible by the prime $p$.
I thought that this proof would be similar to the proof of multiplicity for Euler's totient function, but I have not been able to make the connection. I think once I find a proof for the first part, maybe I will be better able to understand the second part. Any help is appreciated!

Hint:
Use the fact that $\psi(m)$ equals the number of units in the multiplicative group $Z_m^{\times}$. Since $m$ and $n$ are coprime, $
Z_{mn}^{\times}\cong Z_m^{\times}\times Z_n^{\times}$. 
Thus there is $
\psi(mn)=\psi(m)\psi(n)$.
