AIME15 Hardcore TMF user
Number of posts : 163 Age : 21 Location : Pleasanton, CA Registration date : 20080813
 Subject: Simple Proof Wed Aug 13, 2008 8:44 pm  
 Prove that any two odd integers added together make an even integer. 

xoangieexo Starting Member
Number of posts : 7 Age : 22 Registration date : 20080814
 Subject: Re: Simple Proof Thu Aug 14, 2008 6:33 am  
 hmmm well i'm not very sure... but I'll take a guess. Since each odd number is an even integer +1 Two even integers add up to an even integer and 1+1=2. 2+any even integer is even idk if thats how you want me to explain it... 

skittles School Math Nerd
Number of posts : 40 Age : 23 Location : Lost Registration date : 20080813
 Subject: Re: Simple Proof Thu Aug 14, 2008 7:17 am  
 An odd integer is written in the form 2n+1, where n is any integer. 2n+1+2n+1=4n+2. Any multiple of 4 is always even. An even plus an even is always even. Thus, 4n+2 is even.
Pretty much what angie said. 

AIME15 Hardcore TMF user
Number of posts : 163 Age : 21 Location : Pleasanton, CA Registration date : 20080813
 Subject: Re: Simple Proof Thu Aug 14, 2008 1:06 pm  
  skittles wrote:
 An odd integer is written in the form 2n+1, where n is any integer. 2n+1+2n+1=4n+2. Any multiple of 4 is always even. An even plus an even is always even. Thus, 4n+2 is even.
Pretty much what angie said.
You're assuming that both odd integers are the same. The correct proof is: Any odd integer can be written in the form 2n+1, where n is any integer. Another different odd integer can be written as 2k+1. Add these two together to get: 2n+2k+2. This can be factored to 2(n+k+1). Because any 3 integers added together are always an integer, and any integer multiplied by 2 is even, an odd + an odd is ALWAYS even. 

skittles School Math Nerd
Number of posts : 40 Age : 23 Location : Lost Registration date : 20080813
 Subject: Re: Simple Proof Thu Aug 14, 2008 3:02 pm  
 ahahaha wow I missed that.
You could use modular arithmetic too.
Since an odd number is written in the form 2x+1, in mod 2, it'd just be 1 (mod 2) 1 (mod 2) + 1 (mod 2) = 0 (mod 2) Therefore, it is even. 

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 Subject: Re: Simple Proof  
 
