Associated Topics || Dr. Math Home || Search Dr. Math

### Divisor Proof with Contrapositive

```Date: 09/16/2002 at 13:59:01
From: Eric
Subject: Proof of m^2 | n^2 => m | n

I have been trying to prove that if n^2 divides m^2, then n divides m.

m^2 = k*n^2  m,n,k all in Z.

k = (m/n)^2

Is there a proof that says if a rational number squared is an integer,
then it too is an integer? Since we know k is an integer, this would
be sufficent to prove this, correct?  Thank you for any help.

Eric
```

```
Date: 09/16/2002 at 18:08:37
From: Doctor Paul
Subject: Re: Proof of m^2 | n^2 => m | n

I think this proof is most easily accomplished by contraposition.

Can you prove:

n does not divide m => n^2 does not divide m^2

Proving this statement is equivalent to proving your statement, since
a statement and its contrapositive are logically equivalent.

To prove the statement, suppose that n does not divide m. Then there
exists some prime p in the prime factorization of n that does not
occur in the prime factorization of m. If p does not occur in the
prime factorization of m, it won't occur in the prime factorization of
m^2 either. Similarly, if p occurs in the prime factorization of n, it
also occurs in the prime factorization of n^2.

Thus there exists a prime (namely p) in the prime factorization of n^2
that does not occur in the prime factorization of m^2. Hence, n^2 does
not divide m^2.  This was to be shown...

some more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 09/16/2002 at 20:16:47
From: Eric
Subject: Thank you (proof of m^2 | n^2 => m | n)

response it became very clear. Thank you very much!

Eric
```
Associated Topics:
College Logic
High School Logic

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search