For any positive integer n, show the following:(a) ∑d|nσ(d)= ∑d|n(n/d)τ(d)(b) ∑d|n(n/d)σ(d...

For any positive integer n, show the following:

(a) ∑d|nσ(d)= ∑d|n(n/d)τ(d)

(b) ∑d|n(n/d)σ(d)=(∑d|ndτ(d)

[Hint: Because the functions

are both multiplicative, it suffices to prove that F(pk) = G(pk) for any prime p.]