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 Simple Proof

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AIME15
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PostSubject: Simple Proof   Wed Aug 13, 2008 8:44 pm

Prove that any two odd integers added together make an even integer.
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xoangieexo
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PostSubject: 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 Very Happy

idk if thats how you want me to explain it...
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skittles
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PostSubject: 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.
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AIME15
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PostSubject: 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.
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PostSubject: 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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