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# lecture08 - MA1100 Lecture 8 Mathematical Proofs...

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1 MA1100 Lecture 8 Mathematical Proofs Congruences Proof by Cases Absolute Values Chartrand: section 3.4, 4.2, 4.3

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Lecture 8 2 Congruence Definition Let a, b and n be integers with n > 1. If n divides a – b , we say that a is congruent to b modulo n Notation a ª b mod n Example 24 ª 10 mod 7 -2 ª 8 mod 5 84 ª 0 mod 12 Negation a ª b mod n 5 ª 2 mod 6 1 ª 0 mod 4
Lecture 8 3 Congruence Proposition Let a, b and n be integers with n > 1. If a ª b mod n, then a = b + nk for some integer k. Proof Hence a - b = nk for some integer k. Given a ª b mod n . This means n | a – b . i.e. a = b + nk for some integer k. The converse of this proposition is also true

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Lecture 8 4 Congruence a ª b mod n Example ( \$ k œ Z ) ( a = b + nk ) To list integers congruent to b modulo n , We only need to add/subtract multiples of n to/from b. Let b = 3 and n = 4 All integers congruent to 3 modulo 4 : All integers congruent to 0 modulo 4 :
Lecture 8 5 Congruence Proposition Let a and n be integers with n > 1. a ª 0 mod n if and only if n | a Prove this biconditional statement Exercise This gives a criteria for divisibility

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Lecture 8 6 Congruence and Remainder Facts Let a, n be integers with n > 1. Every integer is congruent to exactly one of 0, 1, 2, … , n-1 modulo n. If r is the remainder of a when it is divided by n, then a ª r mod n . Example Division by 5 13 = 5(2) + 3 13 ª 3 mod 5 -13 = 5(-3) + 2 -13 ª 2 mod 5 For division by n, the remainder is between 0 and n-1
Lecture 8 7 Properties of Congruence (1) For every integer a, a ª a mod n (2) For all integers a and b, if a ª b mod n, then b ª a mod n (3) For all integers a, b and c, if a ª b mod n and b ª c mod n, then a ª c mod n reflexive property symmetric property transitive property Proposition Let n be an integer with n > 1.

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Lecture 8 8 Transitive Property of Congruence For all integers a, b, c and n with n > 1, if a ª b mod n and b ª c mod n, then a ª c mod n Proof Proposition Given a ª b mod n .
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lecture08 - MA1100 Lecture 8 Mathematical Proofs...

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