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

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/
```
Associated Topics:
High School Number Theory

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