In regards to perfect and amicable numbers... For any positive integer n, let P*

Question

In regards to perfect and amicable numbers...

For any positive integer n, let P*(n) denote the product of the unitary divisors of n. We say that n is multiplicatively unitary perfect if P*(n)=n^2 and multiplicatively unitary superperfect if P*(P*(n))=n^2. Show that the product of two distinct primes is multiplicative unitary and perfect.

I am offering full points for this answer, so i will require a comprehensive answer with a proof and an explanation of the logic involved in arriving at the conclusion.

Explanation / Answer

Since the number under consideration is the product of 2 distinct prime numbers, the unitary divisors of the number will be the two primes that multiply to give that number and the number itself, i.e,

if n = pq, where p and q are distinct primes, the unitary divisors are p, q and n.

Thus, P*(n) = p*q*n = n^2.

(Since the unitary divisor a number is one which is a divisor and the quotient is coprime with that divisor, the number itself is its own unitary divisor, since 1 is a coprime with all numbers).

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