Product of Two Distinct Primes is Multiplicatively Perfect

Theorem

Let $n \in \Z_{>0}$ be a positive integer which is the product of $2$ distinct primes.

Then $n$ is multiplicatively perfect.

Proof

Let $n = p \times q$ where $p$ and $q$ are primes.

From Product of Two Distinct Primes has 4 Positive Divisors, the positive divisors of $n$ are:

$1, p, q, pq$

Thus the product of all the divisors of $n$ is:

$1 \times p \times q \times p q = p^2 q^2 = n^2$

Hence the result, by definition of multiplicatively perfect.

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $33$