I have this proof of the multiplicativity of the divisor function (I'm actually not sure if that is the correct name for it, since I'm studying it in german). I already know that $d(p_1^{e_1}\cdots p_r^{e_r})=(e_1+1)\cdots(e_r+1)$. Here's the proof:
Let $d:\mathbb{N}\rightarrow\mathbb{N}$ with $$n\mapsto \sum_{d\in\mathbb{N},\:d|n}1$$ be the divisor function.
Let $m=p_1^{e_1}\cdots p_r^{e_r}$ and $n=q_1^{t_1}\cdots q_s^{t_s}$ where $p_i$ and $q_i$ are pairwise diffrent prime numbers. Then the following equalities hold:
$$d(m\cdot n)$$ $$=d\big((p_1^{e_1}\cdots p_r^{e_r})\cdot(q_1^{t_1}\cdots q_s^{t_s})\big)$$ $$=d(p_1^{e_1}\cdots p_r^{e_r}\cdot q_1^{t_1}\cdots q_s^{t_s})$$ $$=(e_1+1)\cdots(e_r+1)\cdot(t_1+1)\cdots(t_s+1)$$ $$=d(p_1^{e_1}\cdots p_r^{e_r})\cdot d(q_1^{t_1}\cdots q_s^{t_s})$$ $$=d(m)\cdot d(n)$$
Can anyone tell me if the proof provided is correct? It seems to be a bit too simple but I can't find any mistakes. Thank you for helping in advance.