lecture19 - Lecture 19 Integers, Division, Primes and...

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Lecture 19 Integers, Division, Primes and Greatest Common Divisors
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Division a divides b c : b = a*c Given: a,b,c integers; a ≠ 0 Notation: a | b
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Division Algorithm Theorem: Let a be an integer, and d be a positive integer. Then there exist unique q and r, with 0 r<d , such that a = dq + r . Note: this is not an algorithm! 101 = 7 14 + 3 -11 = 7 (-2) + 3
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Modular Aritmetic “a is congruent to b modulo m” a b (mod m) a b (mod m) means that a mod m = b mod m m divides (a-b)
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Theorem a b (mod m) k : a = b + km
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Prove or Disprove a,b,c,d,m are all integers a b (mod m) AND c d (mod m) a+c b+d (mod m)
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Prove or Disprove (a+b) mod m = [(a mod m) + (b mod m)] mod m
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Prove or Disprove ac bc (mod m) a b (mod m)
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Applications of Congruences Next, we will briefly go over some practical applications of congruences. Application 1: How to generate random numbers on computers?
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lecture19 - Lecture 19 Integers, Division, Primes and...

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