Prove that all integers are of the form 3k, 3k+1, or 3k+2

The problem is clearly easy and it's obvious why this is true, but I just don't understand how exactly to go about proving it...help?

Thanks!

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- Feb 5th 2010, 05:21 AMeliss93Prove that all integers are of the form...
Prove that all integers are of the form 3k, 3k+1, or 3k+2

The problem is clearly easy and it's obvious why this is true, but I just don't understand how exactly to go about proving it...help?

Thanks! - Feb 5th 2010, 05:36 AMhavaliza
Because the reminder of division any number by 3 is 0, 1 or 2. so $\displaystyle n$ is $\displaystyle 3k+0$ or $\displaystyle 3k+1$ or $\displaystyle 3k+2$

- Feb 5th 2010, 05:37 AMBlack
Follows from the division algorithm (using an integer and 3). For each integer $\displaystyle a$, there exists unique integers $\displaystyle k$ and $\displaystyle r$ such that

$\displaystyle a=3k+r$, where $\displaystyle 0 \le r <3$.