Square of an Odd Number

Date: 11/12/2002 at 09:18:54
From: James Terris
Subject: Divisibility

Is this true or false?

The square of any odd number can be represented in the form 8n+1, 
where n is a whole number.

I think the answer is true, but I can't think of a formula to prove 
it.

Date: 11/12/2002 at 10:11:50
From: Doctor Ian
Subject: Re: Divisibility

Hi James, 

This is interesting! Thanks for asking this question. 

Any odd number can be represented as (2k+1), where k is any integer. 
So the square of an odd number is

  (2k+1)^2 = 4k^2 + 4k + 1

Let's _assume_ that there is some n such that

  4k^2 + 4k + 1 = 8n + 1

Of course, we can subtract 1 from each side to get

      4k^2 + 4k = 8n

And we can divide both sides by 4 to get

        k^2 + k = 2n

         k(k+1) = 2n

So now what does this equation say? It says that if we multiply two
consecutive numbers, we get an even number. Which is true, since one
of those numbers is going to be even. 

Does this help?

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