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I want to prove the following question by contradiction.

Prove that if $x$ is an integer and $3x+2$ is even, then $x$ is even.

Any ideas?

Thank you!

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    Suppose that $3x+2$ is even and $x$ is odd, then $3x$ is odd, and so $3x + 2$should be odd, but it is even, contradiction. – 2017-01-24

1 Answers 1

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  1. Let us assume that $x$ is an odd integer.

  2. Then $3x$ would be odd, since two odd integers multiplied together produce an odd product.

  3. After this, $3x + 2$ will be odd since an odd integer plus an even integer is odd.

  4. But $3x + 2$ is required to be even. This is a contradiction, and so $x$ can not be odd in the first place, so $x$ MUST be even.

Q.E.D.

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    Don't you have to proof why x must be even at the end ? I am a little confused :( – 2017-01-25
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    @AliAziz $x$ is required to be either even or odd. There is no "in between" choice. – 2017-01-25