Checking a proof of a basic property of prime numbers

[Ghost Repeater]

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.

 

[rcgldr]

or p divides both m and n.

 

[member 587159]

or p divides both m and n.

That's not nessecary in the proof. If p divides both m and n, p divides obviously m or n.

 

[PeroK]

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.

You can't conclude that $m/q$ and $n$ are integer factors of $mn/q$.

$q$ might divide neither $m$ nor $n$.

 

[Ray Vickson]

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.

What results do you have available already? What properties are you allowed to use?

 

[Ghost Repeater]

What results do you have available already? What properties are you allowed to use?

The text I'm using has so far defined the greatest common divisor and gives the theorem (with proof) that two nonzero integers a and b have a unique positive greatest common divisor.

I'll try to use these to make the proof. Thanks!

 

[Ghost Repeater]

You can't conclude that $m/q$ and $n$ are integer factors of $mn/q$.

$q$ might divide neither $m$ nor $n$.

I see! Thanks for the heads up!