B11.1 Divisibility Can we equally share n muffins among m persons - PowerPoint PPT Presentation

Discrete Mathematics in Computer Science October 28, 2020 — B11. Divisibility & Modular Arithmetic Discrete Mathematics in Computer Science B11. Divisibility & Modular Arithmetic B11.1 Divisibility Malte Helmert, Gabriele R¨ oger B11.2 Modular Arithmetic University of Basel October 28, 2020 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 1 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 2 / 21 B11. Divisibility & Modular Arithmetic Divisibility B11. Divisibility & Modular Arithmetic Divisibility Divisibility B11.1 Divisibility ◮ Can we equally share n muffins among m persons without cutting a muffin? ◮ If yes then n is a multiple of m and m divides n . ◮ We consider a generalization of this concept to the integers. Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 3 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 4 / 21

B11. Divisibility & Modular Arithmetic Divisibility B11. Divisibility & Modular Arithmetic Divisibility Divisibility Divisibility and Linear Combinations Theorem (Linear combinations) Definition (divisor, multiple) Let a , b and d be integers. If d | a and d | b then Let m , n ∈ Z . If there exists a k ∈ Z such that mk = n , for all integers x and y it holds that d | xa + yb. we say that m divides n , m is a divisor of n or n is a multiple of m and write this as m | n . Proof. If d | a and d | b then there are k , k ′ ∈ Z such that kd = a and k ′ d = b . Which of the following are true? It holds that xa + yb = xkd + yk ′ d = ( xk + yk ′ ) d . ◮ 2 | 4 As x , y , k , k ′ are integers, xk + yk ′ is integer, thus d | xa + yb . ◮ − 2 | 4 Some consequences: ◮ 2 | − 4 ◮ d | a − b iff d | b − a ◮ 4 | 2 ◮ If d | a and d | b then d | a + b and d | a − b . ◮ 3 | 4 ◮ If d | a then d | − 8 a . Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 5 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 6 / 21 B11. Divisibility & Modular Arithmetic Divisibility B11. Divisibility & Modular Arithmetic Divisibility Multiplication and Exponentiation Partial Order If we consider only the natural numbers, divisibility is a partial order: Theorem Theorem Let a , b , c ∈ Z and n ∈ N > 0 . Divisibility | over N 0 is a partial order. If a | b then ac | bc and a n | b n . Proof. Proof. ◮ reflexivity: For all m ∈ N 0 it holds that m · 1 = m , so m | m . If a | b there is a k ∈ Z such that ak = b . ◮ transitivity: If m | n and n | o there are k , k ′ ∈ Z Multiplying both sides with c , we get cak = cb and thus ca | cb . such that mk = n and nk ′ = o . From ak = b , we also get b n = ( ak ) n = a n k n , so a n | b n . With k ′′ = kk ′ it holds then that o = nk ′ = mkk ′ = mk ′′ , and consequently m | o . . . . Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 7 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 8 / 21

B11. Divisibility & Modular Arithmetic Divisibility B11. Divisibility & Modular Arithmetic Modular Arithmetic Partial Order Proof (continued). ◮ antisymmetry: We show that if m | n and n | m then m = n . If m = n = 0, there is nothing to show. Otherwise, at least one of m and n is positive. B11.2 Modular Arithmetic Let this w.l.o.g. (without loss of generality) be m . If m | n and n | m then there are k , k ′ ∈ Z such that mk = n and nk ′ = m . Combining these, we get m = nk ′ = mkk ′ , which implies (with m � = 0) that kk ′ = 1. Since k and k ′ are integers, this implies k = k ′ = 1 or k = k ′ = − 1. As mk = n , m is positive and n is non-negative, we can conclude that k = 1 and m = n . Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 9 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 10 / 21 B11. Divisibility & Modular Arithmetic Modular Arithmetic B11. Divisibility & Modular Arithmetic Modular Arithmetic Halloween is Coming Euclid’s Division Lemma Theorem (Euclid’s division lemma) For all integers a and b with b � = 0 ◮ You have m sweets. there are unique integers q and r ◮ There are k kids showing up for with a = qb + r and 0 ≤ r < | b | . trick-or-treating. Number q is called the quotient and r the remainder. ◮ To keep everything fair, every kid Without proof. gets the same amount of treats. ◮ You may enjoy the rest. :-) Examples: ◮ How much does every kid get, ◮ a = 18 , b = 5 how much do you get? ◮ a = 5 , b = 18 ◮ a = − 18 , b = 5 ◮ a = 18 , b = − 5 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 11 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 12 / 21

B11. Divisibility & Modular Arithmetic Modular Arithmetic B11. Divisibility & Modular Arithmetic Modular Arithmetic Modulo Operation Halloween ◮ With a mod b we refer to the remainder of Euclidean division. ◮ Most programming languages have a built-in operator to compute a mod b (for positive integers): int mod = 34 % 7; // result 6 because 4 * 7 + 6 = 34 def share_sweets(no_kids, no_sweets): ◮ Common application: Determine whether print("Each kid gets", a natural number n is even. no_sweets // no_kids, n % 2 == 0 "of the sweets.") ◮ Languages behave differently with negative operands! print("You may keep", no_sweets % no_kids, "of the sweets.") Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 13 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 14 / 21 B11. Divisibility & Modular Arithmetic Modular Arithmetic B11. Divisibility & Modular Arithmetic Modular Arithmetic Congruence Modulo n Congruence Modulo n – Definition Definition (Congruence modulo n ) For integer n > 1, two integers a and b ◮ We now are no longer interested in the value of the remainder are called congruent modulo n if n | a − b . but will consider numbers a and a ′ as equivalent We write this as a ≡ b (mod n ). if the remainder with division by a given number b is equal. ◮ Consider the clock: Which of the following statements are true? ◮ It’s now 3 o’clock ◮ 0 ≡ 5 (mod 5) ◮ In 12 hours its 3 o’clock ◮ 1 ≡ 6 (mod 5) ◮ Same in 24, 36, 48, . . . hours. ◮ 15:00 and 3:00 are shown the same. ◮ 4 ≡ 14 (mod 5) ◮ In the following, we will express this as 3 ≡ 15 (mod 12) ◮ − 8 ≡ 7 (mod 5) ◮ 2 ≡ − 3 (mod 5) Why is this the same concept as described in the clock example?!? Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 15 / 21 Malte Helmert, Gabriele R¨ oger (University of Basel) Discrete Mathematics in Computer Science October 28, 2020 16 / 21