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**HallsofIvy** Every integer has remainder, when divided by 3, of 0, 1, or 2. It should be easy to see that two numbers, x and y, have x- y divisible by 3 if and only if they have the same remainder when divided by 3. One equivalence class is {0, 3, -3, 6, -6, ...}, the multiples of 3. Another is {1, -2, 4, -5, 7, -8, ...}, numbers that have remainder 1 when divided by 3. The last equivalence class is {2, -1, 5, -4, 8, -7, 11, -10, ...}, numbers that have remainder 2 when divided by 3.

The ordered pairs are any pairs of numbers from one of those three sets.