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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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