# Homework Help: Prove that the square of any integer, when divided by 3. only by odd and even.

1. Nov 6, 2012

### dgamma3

1. The problem statement, all variables and given/known data

I know you could prove this by stating every integer is either 3m, 3m+1 or 3m+2. However I am trying to prove this just using either even numbers or odd numbers.

so for example, when I try:
(2x+1)^2
= 4x^2 + 4x + 1 - expand
= 3x^2 + x^2 + 3x + x + 1 - group like terms
= 3x^2 + 3x + x^2 + x + 1
= 3*(x^2 + x) + x(x+1) + 1

x(x+1) + 1 is an odd number

so
= 3*(x^2 + x) + 2p + 1

thats as far as I can go.

thanks

2. Nov 6, 2012

### Staff: Mentor

What are you trying to prove?
Your thread title seems to be missing a few words
Prove that the square of any integer, when divided by 3, does what?

The problem template has a problem statement section titled, and this is where the statement of the problem should go. Use it.

3. Nov 6, 2012

### dgamma3

sorry mate.

Prove that the square of any integer, when divided by 3, leaves remainder 0 or 1 but never 2.
thanks

4. Nov 6, 2012

### haruspex

What possibilities do you need to consider for the integer?