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Collected Prime Factorization and Multiplicative Functions

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a) State a formula for tau (n), the number of divisions of n, in terms of the collected prime factorization of n.

b) define the term multiplicative function.

c) Suppose that f and g are multiplicative functions. Prove that the function h defined by

h(n) = SUM (d|n) f(d)*g(n/d) is also multiplicative.

d) Find a formula for q(n) in terms of the collected prime factorization for n, where

q(n) = SUM (d/n) tau(d).

e) Find the smallest positive solution to q(n) = 45.

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