Simple Proof

AIME15 · Subject: Simple Proof Wed Aug 13, 2008 7:44 pm

Prove that any two odd integers added together make an even integer.

xoangieexo · Subject: Re: Simple Proof Thu Aug 14, 2008 5:33 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 Very Happy

idk if thats how you want me to explain it...

skittles · Subject: Re: Simple Proof Thu Aug 14, 2008 6:17 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 · Subject: Re: Simple Proof Thu Aug 14, 2008 12:06 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 · Subject: Re: Simple Proof Thu Aug 14, 2008 2:02 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.