Sums of Odd and Even Numbers

Date: 02/04/2002 at 20:30:37
From: Megs
Subject: Odd and Even numbers

I have to prove why the sum of two even numbers is an even number, and 
why the sum of an even and an odd number is an odd number.
 
I've tried using the variable "n" as an odd number, but I don't 
understand where to go from there.  

Thanks.

Date: 02/04/2002 at 22:28:46
From: Doctor Paul
Subject: Re: Odd and Even numbers

Let x and y be even numbers. Then x = 2*k and y = 2*m for some 
integers k and m.

Then 

   x+y = 2*k + 2*m = 2*(k+m) 

which is a multiple of two, and hence is even.

Now suppose x is even but y is odd. Then x = 2*k and y = 2*m + 1 for 
some integers k and m.

Then 

   x+y = 2*k + 2*m + 1 = 2*(k+m) + 1 

which is one more than a multiple of two and hence is odd.

I hope this helps. Please write back if you'd like to talk about this 
more.

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

Date: 02/08/2002 at 23:10:19
From: Megs
Subject: Odd and Even numbers

Your help on my problem helped immensely. Thank you very, very much!

Megan