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Wheel of Congruence Classes of Integers Modulo n

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With respect to a relation , an equivalence class has three properties. Take as an example the integers modulo 3 with respect to congruence.

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1. Reflexivity: . For example, modulo 3, .

2. Symmetry: implies . For example, modulo 3, and .

3. Transitivity: and implies . For example, modulo 3, , and .

The equivalence classes of the integers modulo can be represented as segments of a wheel, with the congruence class of each of the integers shown. Given a positive integer , each integer satisfies the formula , where . Here is the value of the equivalence class (shown over-lined), is the order of the additive group and is the segment, starting from 0 in the innermost circle.

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Contributed by: Theo Williams (October 12)
Open content licensed under CC BY-NC-SA

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