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### 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.

more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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