For a positive integer \(n\), define

\[ F(n)=\sum_{d \mid n} \phi(d) \]

where the sum is over the positive divisors \(d\) of \(n\), including both 1 and n. (For example, the positive divisors of 15 are \(1,3,5\) and 15.)

(a) Calculate \(F(15)\) and \(F(100)\).

(b) Calculate \(F\left(p^{r}\right)\), where \(p\) is prime.

(c) Calculate \(F(p q)\), where \(p, q\) are distinct primes.

(d) Formulate a conjecture about \(F(n)\) for an arbitrary positive integer \(n\). Try to prove your conjecture.