# 18 - LECTURE 18 CONGRUENCE MODULO n Â 8.5 In battling evil...

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Unformatted text preview: LECTURE 18: CONGRUENCE MODULO n ( Â§ 8.5) In battling evil, excess is good; for he who is moderate in announcing the truth is presenting half-truth. He conceals the other half out of fear of the peopleâ€™s wrath. Kahlil Gibran (1883 - 1931) Theorem 1. Let n âˆˆ Z with n â‰¥ 2 . Congruence modulo n is an equivalence relation on Z . Proof 1. Two elements are congruent modulo n if and only if they have the same remainder when dividing by n . So congruence modulo n partitions Z according to remainders. Proof 2. We first show that this relation is reflexive. Let a âˆˆ Z . We want to show a â‰¡ a (mod n ). In other words, we want n | ( a- a ) = 0. This holds since 0 = n Â· 0. Thus a relates to a . We next show that this relation is symmetric. Let a,b âˆˆ Z and assume a â‰¡ b (mod n ). This means n | ( a- b ). In other words a- b = nk for some k âˆˆ Z . We want b â‰¡ a (mod n ), or in other words we want n | ( b- a ). In other words, we want b- a = nâ€˜ for some â€˜ âˆˆ Z . But we have b- a =...
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18 - LECTURE 18 CONGRUENCE MODULO n Â 8.5 In battling evil...

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