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, Bob
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 http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.