# Does this small odd and even proof works?

1. Aug 28, 2013

### Seydlitz

This is taken from Peter J. Eccles, Introduction to Mathematical Reasoning, page 17. This is not a homework because it is an example in the text. Prove that 101 is an odd number. The text has given a way of proving it and I just want to do it with my own approach.

Prove that 101 is an odd number.

Assume 101 is even. There is a number $b$ such that $101=2b$. Adds 1 to both side. $102=2b+1$ The right side shouldn't be divisible by 2 but the left side can be divided by 2. A contradiction? Hence 101 is an odd number.

2. Aug 28, 2013

### Office_Shredder

Staff Emeritus
Given the level of the question it seems like you should prove that 102 can be divided by 2 (by stating what the multiplication is). At that point it's just as easy to say 101 = 50*2+1 so is odd but them's the shakes

3. Aug 28, 2013

### Seydlitz

Well ok then, I just want to know if that contradiction works.

4. Aug 28, 2013

### Tobias Funke

Your proof also uses the fact that if a number isn't even, then it's odd. Again considering the level of the question, this fact may not have been proven yet.

5. Aug 28, 2013

### Seydlitz

It is not proven but it is used in the definition that if a number is not even then it is odd.

6. Aug 29, 2013

### Office_Shredder

Staff Emeritus
Then you should probably think about proving that if a number is of the form 2b+1 then it's odd (in particular you have to prove it's not even)

7. Aug 29, 2013

### Seydlitz

Prove that $2b+1$ is odd.

Suppose $2b+1$ is even, then it exists an integer $c$, where $2b+1=2c$

$2c+1=2b+2$ and $2c-1 = 2b$, hence $2b<2c<2b+2$. Further $b<c<b+1$

But there is no integer which is larger and smaller than the next consecutive integer. So $2b+1$ must be odd.

I think this opens a new bag of cats, but at least I found this myself from scratch!

(The book standard proof implicitly assumes this I believe)

Last edited: Aug 29, 2013
8. Aug 29, 2013

### Office_Shredder

Staff Emeritus
That's a very nice proof

9. Aug 29, 2013

### Seydlitz

I think I can at least feel this small 'beauty' feeling after sketching it. Thanks!