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CS 285- Discrete Mathematics

CS 285- Discrete Mathematics. Lecture 10 . Section 3.4 The Integers & Division. Division Division Theorem Modular Arithmetic Applications of Congruences. Division.

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CS 285- Discrete Mathematics

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  1. CS 285- Discrete Mathematics Lecture 10

  2. Section 3.4 The Integers & Division • Division • Division Theorem • Modular Arithmetic • Applications of Congruences Integers &Division

  3. Division • Definition: If a and b are integers with a ≠ 0, then it is said that a divides b if there is an integer c such that b =ac. • If a divides b, then a is a factor of b and that b is a multiple of a. • Notation: a | b denotes that a divides b. When a doesn’t divide b then, a b • Ex. Determine whether 3 | 7 and whether 3 | 12. Integers &Division

  4. Division Theorems • Let a, b, c be integers. Then • If a | b and a | c, then a | (b +c) • If a | b, then a |bc for all integers c • If a | b and b | c, then a | c • If a | b and a | c, then a | mb + nc (where n & m are integers). Integers &Division

  5. The Division Algorithm • Definition: for an integer a and a positive integer d, there are unique integers q and r with 0 ≤ r < d, such that a = dq +r • Notation: q = a div d, r = a mod d, where • a is the dividend, d is the divisor,q is the quotient, and r is the remainder. • Ex. What is the quotient and remainder when 101 is divided by 11. 101 = 11 * 9 + 2 Hence the quotient is 101 div 11 = 9, and remainder is 101 mod 11 = 2 Integers &Division

  6. Modular Arithmetic • If a and b are integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if a mod m ≡ b (mod m). thus m divides a – b. • Let m be a positive integer. The integers a and b are congruent modulo m if and only if there is an integer k such that a == b + km. • Let m be a positive integer. If a ≡ b (mod m) and c ≡ d (mod m) then a + c ≡ b + d (mod m) and ac = bd (mod m) • Ex. 7 ≡ 2 (mod 5) and 11 ≡ 1 (mod 5) then 18 = 7 + 11 = 2 + 1 = 3 (mod 5) 77 = 7 * 11 = 2 * 1 = 2 (mod 5) Integers &Division

  7. Modular arithmetic (Cont.) • Let m be a positive integer. And let a and b be integers. Then ( a + b) mod m = ((a mod m) + (b mod m)) mod m ab mod m = ((a mod m)(b mod m)) mod m Ex. (3 + 5)= 8 mod 2 =0 = ((3 mod 2) + ( 5 mod 2 ) ) mod 2 = ( 1 + 1 ) mod 2 = 0 Integers &Division

  8. Hashing function • H(k) = k mod m, where k and m are positive integers. • Hashing is used to assign available memory locations for saving files. • Ex. Let m = 111 • H(06212848) = 06212848 mod 111 = 14 • H(037149212)=037149212 mod 111 = 65 Integers &Division

  9. Pseudorandom Numbers • Randomly chosen numbers are generated by using the congruence xn+1 = (ax0 + c) mod m Where the four integers are: the modulus m, the multiplier a, increment c and the seed x0 Ex. m = 9, a =7, c = 4 and x0 = 3 x1 = 7x0 + 4 mod 9 = 21 + 4 mod 9 = 25 mod 9 = 7 x2 = 7x1 + 4 mod 9 = 49 + 4 mod 9 = 53 mod 9 = 8 x3 = 7x2 + 4 mod 9 = 56 + 4 mod 9 = 60 mod 9 = 6 x4 = 7x3 + 4 mod 9 = 42 + 4 mod 9 = 46 mod 9 = 1 Thus the sequence generated is : 3, 7, 8, 6, 1, ….. Integers &Division

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