PMATH340 LN8

Suppose that a and b have the same nonnegative

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Unformatted text preview: Now n|(n(q − q ￿ ) + (r − r￿ )), which says that n|(r − r￿ ) Since both r, r￿ < n this gives r − r￿ = 0, so that r = r￿ . Suppose that a and b have the same nonnegative remainder when divided by n, say r. So a − b = qn + r − (q ￿ n + r) = (q − q ￿ )n, which gives n|(a − b) and so a ≡ b(mod n). Theorem 9. Let n ∈ N and a, b, c ∈ Z. Then (i) a ≡ a(mod n) (reflexive) (ii) if a ≡ b(mod n), then b ≡ a(mod n) (symmetric) (ii) if a ≡ b(mod n) and b ≡ c(mod n), then a ≡ c(mod n) (transitiv...
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This note was uploaded on 02/09/2013 for the course PMATH 340 taught by Professor W.alabama during the Spring '09 term at Waterloo.

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