Prove: The sum of two even integers is even.

[jcink]

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?

[stapel_eliz]

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.

2. Why the 2 at all?

What makes a number "even"?

3. What exactly is m and n?

They are variables.

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"?)

[jcink]

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.

[stapel_eliz]

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.

[jcink]

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.