# Answer on Abstract Algebra Question 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 n

n

=p1*p1*p1*p2*p2*p2* … *

*pk*pk*pk

Hence if p divides n

Similarly for q.

Since p and q are distinct numbers, it follows that

(pq)

also divides n

Let

n= p1 * p2 * … * pk,

be such a presentation. Some of multiples may repeat.

Then n

^{3}has the following presentationn

^{3}= p1^{3}* p2^{3}* … * pk^{3}==p1*p1*p1*p2*p2*p2* … *

*pk*pk*pk

Hence if p divides n

^{3}, then it divides n, and therefore p^{3}divides n^{3.}Similarly for q.

Since p and q are distinct numbers, it follows that

(pq)

^{3}= p^{3}q^{3}also divides n

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