$ \sum_{a \mid n} \Bigl( f(a) \sum_{b \mid \frac{n}{a}} g(b) \Bigr) = \sum_{a \mid n} \Bigl( g(a) \sum_{b \mid \frac{n}{a}} f(b) \Bigr) $ Is the following equation true?
$$
\sum_{a \mid n} \Bigl( f(a) \sum_{b \mid \frac{n}{a}} g(b) \Bigr) =
\sum_{a \mid n} \Bigl( g(a) \sum_{b \mid \frac{n}{a}} f(b) \Bigr)
$$
If this equation is true, how can I prove it?

Hint: both sides are equal to $\displaystyle\sum_{\substack{a,b,c\\abc=n}}f(a)g(b)$.
