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Ghost Repeater

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## Homework Statement

Prove: If p is prime and m, n are positive integers such that p divides mn, then either p divides n or p divides m.

Is anyone willing to look through this proof and give me comments on the following: a) my reasoning within the strategy I chose (validity, any constraints or cases I might have missed), b) the strategy I chose (was it a good one, is there a better one, some other angle I might have taken), and c) my presentation (messiness, inelegance)?

## Homework Equations

[/B]n/a## The Attempt at a Solution

Here's my proof.

If p divides mn, then there must be some positive integer q such that mn = pq. Then (mn)/q = p is prime. The factors of p are m/q and n. Since p is prime, one of these factors must be equal to 1. Therefore either

a) n = 1

or

b) m/q = 1.

If a) holds, then mn = m = pq and we see that p divides m.

If b) holds, then m = q, and so mn = qn = pq, which means p = n and so p divides n (with quotient of 1).

Therefore, if p divides mn, then either p divides m or p divides n, as desired.