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Discrete Mathematics

Discrete Mathematics. 03.20.09. Review. Division Algorithm a = dq + r Greatest Common Divisor (GCD) GCD( a,b ) – the largest integer that divides both a and b Least Common Multiples (LCM) LCM( a,b ) – the smallest positive integer that is divisible by both a and b. Review.

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Discrete Mathematics

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  1. Discrete Mathematics 03.20.09

  2. Review • Division Algorithm • a = dq + r • Greatest Common Divisor (GCD) • GCD(a,b) – the largest integer that divides both a and b • Least Common Multiples (LCM) • LCM(a,b) – the smallest positive integer that is divisible by both a and b

  3. Review • Prime • A positive integer greater than 1 with exactly two positive integer divisors • Relatively Prime Integers • Integers a and b such that GCD(a,b) = 1 • Pairwise Relatively Prime • A set of integers with the property that every pair of these integers is relatively prime

  4. Today’s Topics • Modular Arithmetic • Applications of Modular Arithmetic

  5. Modular Arithmetic • In some situations, we care only about the remainder of an integer when it is divided by some specified positive integer. • Ex.: Identifying if an integer is positive or negative.

  6. Congruences • If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a – b. • a b (mod m) if m | a - b • Definition of Notations: • a b (mod m) • a is congruent to b modulo m • a b (mod m) • a is not congruent to b modulo m • m | a – b • m divides a - b /

  7. Example • Determine whether 17 is congruent to 5 modulo 6. • Determine whether 24 and 14 are congruent to modulo 6.

  8. Exercise • Decide whether each of these integers is congruent to 5 modulo 17. • 80 • 103 • - 29 • - 122 • 35

  9. Applying Modular Arithmetic • Problem 1: • What time will it be 50 hours from now?

  10. Applying Modular Arithmetic • Problem 2: • Generating pseudorandom numbers generated by choosing m=9, a=7, c=4 and x0=3. • Find: • xn+1 = (axn + c) mod m • Find • x1 , x2, x3, x4, x5, x6, x7, x8, x9

  11. Applying Modular Arithmetic • Problem 3: • Cryptology • Encrypt the word HELLO using f(p) = p+3

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