33 theorem 7 the set 0 1 2 n 1 is a complete

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Unformatted text preview: r each integer k there corresponds an ai such that k ≡ ai (mod n). 33 Theorem 7. The set {0, 1, 2, . . . , n − 1} is a complete residue system modulo n. Proof. This is a direct consequence of the Division Algorithm. Theorem 8. Let a, b ∈ Z, then a ≡ b(mod n) if and only if a and b have the same nonnegative remainder when divided by n. Proof. Let a, b ∈ Z. Suppose a ≡ b(mod n). Then by definition n|(a − b). Now by the Division Algorithm there are q, q ￿ , r, r￿ ∈ Z with 0 ≤ r, r￿ < n so that a = qn + r and b = q ￿ n + r￿ ....
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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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