For any positive integer N, the function f(N) is defined by

where are the only prime numbers that are factors of N.

Thus

(1) Evaluate f(12) and f(180)

(2) Show that f(N) is an integer for all N

(3) Prove, or disprove by means of a counterexample, each of the following:

(a)

(b) if p and q are distinct prime numbers;

(c) only if p and q are distinct prime numbers.

(4) Find a positive integer m and a prime number p such that