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Prove x^2+y^2 Not Divisible by 4

Date: 09/20/2001 at 14:22:59
From: Robert Breiner
Subject: Prove x^2+y^2 is not divisible by 4

Dr. Math,

I have attempted to prove this problem several times but have not been 
able to get it right. I am not even sure how to start it correctly. 
Any assistance would help a great deal.

Prove that if x and y are odd, then x^2 + y^2 is even but not 
divisible by 4.

Thanking you,

Date: 09/20/2001 at 17:14:28
From: Doctor Jubal
Subject: Re: Prove x^2+y^2 is not divisible by 4

Hi Robert,

Thanks for writing to Dr. Math.

Keeping track of the fact that x and y are odd can sometimes be a bit
troublesome, so it's best if we can write them in such a way that they 
are inherently odd without us having to remember it.

I suggest writing them as 2a+1 and 2b+1, where a and b could be any 
natural number, odd or even. Then we could proceed by  writing 
x^2 + y^2 as

  (2a+1)^2 + (2b+1)^2

  (4a^2 + 4a + 1) + (4b^2 + 4b +1)

  4(a^2 + b^2 + a + b) + 2

So x^2 + y^2 is 2 more than a multiple of 4. Therefore it must be 
even, but can never be a multiple of 4 itself.  

See if you can prove this: if x and y are even numbers not divisible 
by 4, then x^2 + y^2 must be divisible by 8, but not 16.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum   
Associated Topics:
High School Number Theory

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