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The sum of two even integers is even, the sum of an even and an odd integer is odd, and the sum of two odd integers is even. What is the generalization of this statement to residue classes mod 3?

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The generalization is an addition table mod n.
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An even number can be written as 2n+0 and an odd number can be written 2n+1

Apply this idea to the number 3 and you get 3n+0, 3n+1 and 3n+2. You can show how addition works by adding two of them together and doing a little bit of algebra.

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