# Prove that all integers are of the form...

• Feb 5th 2010, 06:21 AM
eliss93
Prove 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, 06:36 AM
havaliza
Because the reminder of division any number by 3 is 0, 1 or 2. so $n$ is $3k+0$ or $3k+1$ or $3k+2$
• Feb 5th 2010, 06:37 AM
Black
Follows from the division algorithm (using an integer and 3). For each integer $a$, there exists unique integers $k$ and $r$ such that

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