Proof that an Even Number Squared is Even

Date: 06/02/99 at 21:03:21
From: Jason 
Subject: Proof: Even number squared is even

How do you prove that any even number squared is even and any odd 
number squared is odd?

For the odd, I let n be the odd number. Then say that n = 2k + 1, so 
n^2 = 4k^2 + 4k + 1, but from here I am not sure where to go.

For the even number I have not gotten very far.

Thanks,
Jason

Date: 06/03/99 at 08:27:12
From: Doctor Rick
Subject: Re: Proof: Even number squared is even

Hi, Jason.

You're practically done. You can factor a 2 out of the first two terms 
of your expansion:

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

Since m = 2k^2 + 2k is an integer, you have expressed n^2 as 2m+1, 
making it odd.

The even case is easier: if n = 2k, then n^2 = 4k^2 = 2(2k^2), which 
is 2 times an integer and therefore is even.

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