Answer to Question #2579 in Abstract Algebra for Josh Haver

Question

Prove that if two distinct primes p and q divide n^3, then n^3 is a multiple of (pq)^3.

Answer 1

The number n can be uniquely represented as a product of primes up to their permutation.
Let
n= p1 * p2 * … * pk,
be such a presentation. Some of multiples may repeat.
Then n³ has the following presentation
n³= p1³ * p2³ * … * pk³=
=p1*p1*p1*p2*p2*p2* … *
*pk*pk*pk

Hence if p divides n³, then it divides n, and therefore p³ divides n^3.
Similarly for q.
Since p and q are distinct numbers, it follows that
(pq)³ = p³ q³
also divides n³