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My tutor gave an online assignment with this question: Proof that 8 is an even number by using contradiction technique

I've read about Proof by contradiction but I couldn't come up with any idea on how to start solving this.

Any help is much appreciated. Thanks.

khalid redza
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    "Suppose not" is the best first sentence. Then if 8 is not even it must be odd, right? Then by definition... (work from here) – Sean Roberson Sep 25 '16 at 00:16
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    Assume its not even. $8=2\cdot 4$ is even. Contradiction. Thus $8$ is even. – Rene Schipperus Sep 25 '16 at 00:22
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    I find the assignment quite odd to be honest, since the direct proof is so more natural in this case. That said: suppose $8$ is odd, then $8-2$ will be odd as well, and so will be $8-4$ and $8-6=2$. But $2= 2 \cdot 1$ is provably even, so that's a contradiction. – dxiv Sep 25 '16 at 00:38
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    If you have a personal tutor, why are you asking the question on stackexchange? Your tutor could equally give you the answer but felt you would learn more by actually working it out yourself. In this case, the tutor likely gave you the simplest possible thing ("prove 8 is even"). Given the proof structure that @SeanRoberson reminds you about, was there really any reason you needed the additional simpleart answer to fill in the last step (using the definition of "even")? I think both your tutor and Sean helped you more, yet you did not bother to thank them (as you did simpleart). – Michael Sep 25 '16 at 02:27
  • I am so sorry, I want to apologize to Sean Roberson, Rene Schipperus, divx and Michael. I want to thank all of you for helping me with the question. It's just that I had a class after that and just now I have the free time to check again. My tutor was giving the assignment before he realised that he didn't give out the notes in our emails. That's why I had to come here to ask for help. I am so sorry if I offended anyone. – khalid redza Sep 25 '16 at 02:57
  • What is your definition of $8$? – Christian Blatter Sep 25 '16 at 08:08
  • :) Cheers. ${}$ – Simply Beautiful Art Sep 25 '16 at 09:45

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Assume it is not even. Then it must be odd. If it is odd, it isn't divisible by $2$. But $8÷2=4$, so $8$ is divisible by $2$. This is in contradiction to the statement that it must be odd. Thus, it must be even.

Simply Beautiful Art
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