Dirichlet Convolution. If $f,g:\mathbb {N} \to \mathbb {C}$ are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution $f ∗ g$ is a new arithmetic function defined by: $$(f*g)(n) \ =\ \sum_{d\,\mid \,n} f(d)\,g\!\left(\frac{n}{d}\right) $$
Questions
We denote $C_n$ the set of pairs $(d1,d2)$ of $N$ such that $d_1d_2 = n$. Prove that : $(f * g)(n)=∑(d1,d2)\in C_n f(d1)g(d2)$, then conclude that "$*$" is commutative.
We denote $Γ_n$ the set of triplets $(d1,d2,d3)$ of $N$ such that $d_1d_2d_3=n.$ Prove that : $((f * g) * h)(n)=∑(d1,d2,d3)∈Γ_n f(d1)g(d2)h(d3)$, then conclude that "$*$" is associative.
I've been able to solve the first quest through proving that the application : $d\to (d,n/d)$ is one to one, where $d$ being a divisor of $n$, and $(d,n/d)$ in $C_n)$
I'm stuck in the second qst please any help.