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Answer to Question #2579 in Abstract Algebra for Josh Haver

Question #2579
Prove that if two distinct primes p and q divide n^3, then n^3 is a multiple of (pq)^3.
Expert's answer
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

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