# Prove that 2x is always an even number.

How does one prove that for every value in $\Bbb N$ 2x = an even number?

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Define an even number –  Belgi Sep 14 '12 at 10:55

Isn't the definition of an even number all $n \in \mathbb{Z}$ such that n = 2k for some $k \in \mathbb{Z}$?

$2x$ with $x \in \mathbb{N}$ would always be an even number according to the definition above.

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Basically, if someone told you to prove the above statement, then you would simply write, "by definition" and maybe even invoke the statement of the definition. But even that would be excessive in my opinion. –  aviness Sep 14 '12 at 11:40
That's the definition of an even number. A natural number $y$ is said to be even if there is another natural number $x$ such that $y = 2x$
Another definition: A natural number $n$ is said to be even if its residue class in the quotient ring (field) $\Bbb{Z}/2\Bbb{Z}$ is not different from 0.