# Does this small odd and even proof works?

by Seydlitz
Tags: proof
 P: 234 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.
 Mentor P: 4,499 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
P: 234
 Quote by Office_Shredder 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
Well ok then, I just want to know if that contradiction works.

P: 134

## Does this small odd and even proof works?

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.
P: 234
 Quote by 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.
It is not proven but it is used in the definition that if a number is not even then it is odd.
Mentor
P: 4,499
 Quote by Seydlitz It is not proven but it is used in the definition that if a number is not even then it is odd.
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)
 P: 234 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
 Mentor P: 4,499 That's a very nice proof
P: 234
 Quote by Office_Shredder That's a very nice proof
I think I can at least feel this small 'beauty' feeling after sketching it. Thanks!

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