If n^2 is Even, n is Even

Date: 02/21/2002 at 23:42:26
From: Mary Jackson
Subject: Show that n^2 is even then n is even.

I have to show that if n^2 is even then n is also even. I don't know 
where to start. I would love your help. Thanks.

Date: 02/22/2002 at 00:45:20
From: Doctor Paul
Subject: Re: Show that n^2 is even then n is even.

Try a proof by contraposition (are you familiar with this method of 
proof? a statement and its contrapositive are logically equivalent so 
proving the contrapositive of the original statement actually proves 
the original statement).  

So what you want to do is assume that n is not even (i.e., it is odd) 
and show that n^2 is not even (i.e., it is odd).

n is odd means that you can write n = 2*k + 1 for some integer k. Then 

   n^2 = 4*k^2 + 4*k + 1 

       = 2 * (2*k^2 + 2*k) + 1 

which is clearly odd.  

This completes the proof.

I hope this helps.  Please write back if you'd like to talk about this 
more.

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