Answer to Question #2579 in Abstract Algebra for Josh Haver
Prove that if two distinct primes p and q divide n^3, then n^3 is a multiple of (pq)^3.
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 n3. Similarly for q. Since p and q are distinct numbers, it follows that (pq)3 = p3 q3 also divides n3