Prove: The sum of two even integers is even.  TOPIC_SOLVED

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Prove: The sum of two even integers is even.  TOPIC_SOLVED

Postby jcink on Sat Oct 05, 2013 6:29 am

Hello, I don't understand how this is "proven" and if someone could please break this down further for me and "explain it like I'm 5" so to speak that would be much appreciated.

Problem: Prove: The sum of two even integers is even. Use a Direct Proof.

The Proof:

Code: Select all
a + b even integers
a = 2n
b = 2m

a + b = 2n + 2m
a + b = 2(n + m)


Supposedly it is now proven. I do not get this at all, here are some questions:

1. Why do we set a and b to 2n and 2m?
2. Why the 2 at all?
3. What exactly is m and n?
4. How does 2(n + m) prove this question true at all? For example, how can I do a check using the proof to see some true outcomes?
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Postby stapel_eliz on Sat Oct 05, 2013 12:50 pm

jcink wrote:1. Why do we set a and b to 2n and 2m?

To express the numbers in a generic way, but in a way that shows that the numbers are even, not odd. This is the same process that you used back in beginning algebra when you did "number" word problems.

jcink wrote:2. Why the 2 at all?

What makes a number "even"?

jcink wrote:3. What exactly is m and n?

They are variables.

jcink wrote:4. How does 2(n + m) prove this question true at all? For example, how can I do a check using the proof to see some true outcomes?

Think about what it means for a number to be "even". (Hint: What must be a factor of the number, for it to be "even"?) :wink:
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Re: Prove: The sum of two even integers is even.

Postby jcink on Sat Oct 05, 2013 8:24 pm

Thanks for the reply, that really helped me out.

So we're basically setting variables and using 2 * a number to ensure it is even for our proof, then factoring to simplify it.
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Postby stapel_eliz on Sat Oct 05, 2013 11:27 pm

The definition of an "even" number is that it is a multiple of two. Therefore, the generic form of an even number is the product of two and some variable. To show that the sum is also even, it must be shown that the sum is a multiple of two. :wink:
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Re: Prove: The sum of two even integers is even.

Postby jcink on Sat Oct 05, 2013 11:56 pm

Makes a lot of sense now. Thanks. I think I was having a brain fart.

By the way thanks a lot for purple math. I've been using it since middle school and it's always been a great resource for math. I never noticed until recently that there was a forum.
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