2. Module #9 – Number Theory 6/11/20161 Chapter 3 Algorithms, Integers and Matrices

3. Module #9 – Number Theory 6/11/20162 Section 3.4: The Integers and Division DivisionDivision Let a,b  Z with a  0.Let a,b  Z with a  0. a|b  “a divides b” :  “  c  Z: b=ac” So: a is a factor or a divisor of b, and b is a multiple of a.a|b  “a divides b” :  “  c  Z: b=ac” So: a is a factor or a divisor of b, and b is a multiple of a. a doesn’t divide b is denoted by a | b.a doesn’t divide b is denoted by a | b. –Example: 3   12  True, but 3  7  False.

4. Module #9 – Number Theory 6/11/20163 Facts: the Divides Relation  a,b,c  Z:  a,b,c  Z: 1. (a|b  a|c)  a | (b + c) 1. (a|b  a|c)  a | (b + c) 2. a|b  a|bc 3. (a|b  b|c)  a|c Examples: 1. 3  12  3  9  3  (12+9)  3  21 (21 ÷ 3=7) 1. 3  12  3  9  3  (12+9)  3  21 (21 ÷ 3=7) 2. 2  6  2  (6×3)  2  18 (18 ÷2= 9) 2. 2  6  2  (6×3)  2  18 (18 ÷2= 9) 3. 4  8  8  64  4  64 (64 ÷ 4 = 16 ) 3. 4  8  8  64  4  64 (64 ÷ 4 = 16 )

5. Module #9 – Number Theory 6/11/20164 Composite and Prime Numbers A positive integer p>1 is prime if the only positive factors of p are 1 and pA positive integer p>1 is prime if the only positive factors of p are 1 and p ☻ Some primes: 2,3,5,7,11,13... Non-prime integers greater than 1 are called composite, because they can be composed by multiplying two integers greater than 1.Non-prime integers greater than 1 are called composite, because they can be composed by multiplying two integers greater than 1.

6. Module #9 – Number Theory 6/11/20165 Fundamental Theorem of Arithmetic Every positive integer greater than 1 has a unique representation as a prime or as the product of two or more primes where the prime factors are written in order of non- decreasing size. (tree or division)Every positive integer greater than 1 has a unique representation as a prime or as the product of two or more primes where the prime factors are written in order of non- decreasing size. (tree or division) 100 = 2·2·5·5= 2 2 5 2 100 = 2·2·5·5= 2 2 5 2 13 = 13 13 = 13 1024 = 2·2·2·2·2·2·2·2·2·2=2 10

7. Module #9 – Number Theory 6/11/20166 Theorem If n is composite integer, then n has prime divisorIf n is composite integer, then n has prime divisor ≤ Ex: 49  prime numbers less than are 2,3,5,7Ex: 49  prime numbers less than are 2,3,5,7 16  prime numbers less than are 2,3 16  prime numbers less than are 2,3 An integer n is prime if it is not divisible by any prime ≤An integer n is prime if it is not divisible by any prime ≤ Ex: 13 where = 3.6 so the prime numbers are 2,3 but non of them divides 13 so 13 is primeEx: 13 where = 3.6 so the prime numbers are 2,3 but non of them divides 13 so 13 is prime

8. Module #9 – Number Theory 6/11/20167 To find the prime factor of an integer n:To find the prime factor of an integer n: 1- find 2- list all primes ≤ 2, 3, 5, 7,…root of n 2, 3, 5, 7,…root of n 3- find all prime factors that divides n.

9. Module #9 – Number Theory 6/11/20168

10. Module #9 – Number Theory 6/11/20169 The Division “Algorithm” let a be an integer and d a positive integer, then there are unique integers q and r such that:a = d × q + r, 0  r < dlet a be an integer and d a positive integer, then there are unique integers q and r such that:a = d × q + r, 0  r < d d is called divisor, and a is called dividendd is called divisor, and a is called dividend q is the quotient, and r is the remainder (positive integer)q is the quotient, and r is the remainder (positive integer)  q = a div d, r =a mod d

11. Module #9 – Number Theory 6/11/201610 Examples What are the quotient and remainder when 101 is divided by 11?What are the quotient and remainder when 101 is divided by 11? 101 = 11 × 9 + 2, q = 9, r = 2 101 = 11 × 9 + 2, q = 9, r = 2 What are the quotient and the remainder when -11 is divided by 3?What are the quotient and the remainder when -11 is divided by 3? -11 = 3 × (-4) + 1,q = -4, r = 1-11 = 3 × (-4) + 1,q = -4, r = 1 Note: -11  3 × (-3) - 2 because r  0 (can't be negative)Note: -11  3 × (-3) - 2 because r  0 (can't be negative)

12. Module #9 – Number Theory 6/11/201611 Modular Congruence Let a, b  Z, m  Z +. Thm: a is congruent to b modulo m a is congruent to b modulo m written “a  b (mod m)”, iff 1) a mod m = b mod m 1) a mod m = b mod m 2) m | a  b i.e. (a  b) mod m = 0. 2) m | a  b i.e. (a  b) mod m = 0.

13. Module #9 – Number Theory 6/11/201612 Examples Is 17 congruent to 5 modulo 6 ? 17 mod 6 = 5 and5 mod 6 = 5 17 mod 6 = 5 and5 mod 6 = 5 and 6 | (17-5)  6 | 12 where 12 ÷ 6 = 2 Then 17 ≡ 5 (mod 6) Is 24 congruent to 14 modulo 6? Since, 24 mod 6 = 0 and14 mod 6 = 2 Then, 24 ≡ 14 (mod 6) and 6 | (24-14)  6 | 10

14. Module #9 – Number Theory 6/11/201613 Useful Congruence Theorems Let a,b,c,d  Z, m  Z +. Then if a  b (mod m) and c  d (mod m), then:Let a,b,c,d  Z, m  Z +. Then if a  b (mod m) and c  d (mod m), then: ▪ a+c  b+d (mod m), and ▪ ac  bd (mod m)

15. Module #9 – Number Theory 6/11/201614 Ex. : let 7 ≡ 2 (mod 5) and let 7 ≡ 2 (mod 5) and 11 ≡ 1 (mod 5) 11 ≡ 1 (mod 5)Then: (7 + 11) ≡ (2 + 1) (mod 5)  18 ≡ 3 (mod 5) (7 + 11) ≡ (2 + 1) (mod 5)  18 ≡ 3 (mod 5) and and ( 7 × 11) ≡ (2 × 1) (mod 5)  77 ≡ 2 (mod 5) ( 7 × 11) ≡ (2 × 1) (mod 5)  77 ≡ 2 (mod 5)

16. Module #9 – Number Theory 6/11/201615 Greatest Common Divisor (GCD) The greatest common divisor gcd(a,b) of integers a,b (not both 0) is the largest (most positive) integer d that is a divisor both of a and of b.The greatest common divisor gcd(a,b) of integers a,b (not both 0) is the largest (most positive) integer d that is a divisor both of a and of b.

17. Module #9 – Number Theory 6/11/201616 Ways to find GCD 1.find all positive common divisors of both a and b, then take the largest divisor Ex: find gcd (24, 36)? Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18,36 Common divisors: 1, 2, 3, 4, 6, 12 MAXIMUM = 12  gcd (24, 36) = 12

18. Module #9 – Number Theory 6/11/201617 2. use prime factorization: Take the min power of common factors Take the min power of common factorsExample Find gcd (24, 36) Find gcd (24, 36) 24 = 2 × 2 × 2 × 3 = 2 3 × 3 24 = 2 × 2 × 2 × 3 = 2 3 × 3 36 = 2 × 2 × 3 × 3 = 2 2 × 3 2 36 = 2 × 2 × 3 × 3 = 2 2 × 3 2 gcd(24,36) = 2 2 × 3=12 gcd(24,36) = 2 2 × 3=12

19. Module #9 – Number Theory 6/11/201618 Ex: find gcd (120, 500)? Ex: find gcd (120, 500)? 120 = 2 3 × 3 × 5 500 = 2 2 × 5 3  gcd (120, 500) = 2 2 × 5 = 20 Ex: find gcd (17, 22)? No common divisors so gcd (17, 22) = 1 so, the numbers 17 and 22 are called relatively prime.

20. Module #9 – Number Theory 6/11/201619 Least Common Multiple lcm(a,b) of positive integers a, b, is the smallest positive integer that is a multiple both of a and of b.lcm(a,b) of positive integers a, b, is the smallest positive integer that is a multiple both of a and of b.

21. Module #9 – Number Theory 6/11/201620 Find lcm(6,10) ? Take the max power of all Factors 6 = 2×3 10 = 2×5 10 = 2×5 Lcm(6,10) = 2×3×5=30 Lcm(6,10) = 2×3×5=30 Find Lcm (24, 36)? 24 = 2 3 × 3 1 36 = 2 2 × 3 2  Lcm (24, 36) = 2 3 × 3 2 = 72

22. Module #9 – Number Theory 6/11/201621 Section 3.1: Algorithms A finite set of instructions for performing a computation or for solving a problemA finite set of instructions for performing a computation or for solving a problemExamples: -Finding the largest integer in a finite sequence of integers -Locating an element in a finite set -Sorting elements in a finite sequence

23. Module #9 – Number Theory 6/11/201622 Finding the maximum element procedure max(a 1,a 2,……..,a n : integers) max:=a 1 for i:=2 to n if max<a i then max:=a i if max<a i then max:=a i {max is the largest element}

24. Module #9 – Number Theory 6/11/201623 Linear Search Algorithm procedure linearsearch (x:integer, a 1,a 2,..,a n :integers) i:=1 While(i≤n and x ≠ a i ) i = i+1 i = i+1 If i ≤ n then location:= i else location:= 0 {location is the subscript of the term that equals x or the location is 0 if x is not found }

25. Module #9 – Number Theory 6/11/201624 Bubble Sort Algorithm procedure bubblesort(a 1,a 2,…..,a n ) for i:=1 to n-1 for j:=1 to n-i for j:=1 to n-i if a j > a j+1 then interchange a j and a j+1 if a j > a j+1 then interchange a j and a j+1 {a1,…..,an is in increasing order}

26. Module #9 – Number Theory 3.6 Integers and Algorithms Introduction: The term algorithm originally referred to procedures for performing arithmetic operations using the decimal representations of integers. These algorithms, adapted for use with binary representations, are the basis for computer arithmetic. 6/11/201625

27. Module #9 – Number Theory There are many important algorithms involving integers besides those used in arithmetic, We will describe an algorithm for finding the base b expansion of a positive integer for any base b. 26

28. Module #9 – Number Theory Representations of Integers In everyday life we use decimal notation to In everyday life we use decimal notation to express integers. express integers. For example, 965 is used to Denote 9·10 2 + 6·10 + 5. 9·10 2 + 6·10 + 5. However, it is often convenient to use bases other than 10. 27

29. Module #9 – Number Theory Computers usually use binary notation Computers usually use binary notation (with 2 as the base) when carrying out arithmetic, (with 2 as the base) when carrying out arithmetic, and octal (base 8) or hexadecimal (base 16) and octal (base 8) or hexadecimal (base 16) notation when expressing characters, such as notation when expressing characters, such as letters or digits. In fact, we can use any letters or digits. In fact, we can use any positive integer greater than 1 as the base positive integer greater than 1 as the base when expressing integers. when expressing integers. 6/11/201628

30. Module #9 – Number Theory BINARY EXPANSIONS What is the decimal expansion of the integer that has (l 0101 1111) 2 as its binary expansion? Solution: We have (l 0101 111l) 2 = 1 · 2 8 + 0 · 2 7 + 1 · 2 6 + 0 · 2 5 + 1 · 2 4 + 1 · 2 3 + 1 · 2 2 + 1 · 2 1 + 1 · 2 0 + 1 · 2 3 + 1 · 2 2 + 1 · 2 1 + 1 · 2 0 = 351. = 351. 6/11/201629

31. Module #9 – Number Theory HEXADECIMAL EXPANSIONS Sixteen is another base used in computer science. The base 16 expansion of an integer is called its hexadecimal expansion. hexadecimal digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F, where the letters A through F represent the digits corresponding to the numbers 10 through 15 (in decimal notation). (in decimal notation). 6/11/201630

32. Module #9 – Number Theory 6/11/201631 EXAMPLE 2 What is the decimal expansion of the hexadecimal expansion of (2AE0B) 16 ? Solution: We have (2AE0B) 16 = 2.16 4 +10.16 3 +14.16 2 +0·16 +11 = ( 175627) 10. = ( 175627) 10.

33. Module #9 – Number Theory EXAMPLE 1: Find the binary expansion of (241) 10 Solution: First divide 241 by 2 to obtain 241 = 2. 120 + 1. Successively dividing quotients by 2 gives 120 = 2 · 60 + 0 60 = 2 · 30 + 0 30 = 2 · 15 + 0 15 = 2 · 7 + 1 7 = 2 · 3 + 1 7 = 2 · 3 + 1 3 = 2 · 1 + 1 3 = 2 · 1 + 1 1 = 2 · 0 + 1. Therefore, (241) 10 = (1111 0001) 2 1 = 2 · 0 + 1. Therefore, (241) 10 = (1111 0001) 2 6/11/201632 BASE CONVERSION

34. Module #9 – Number Theory EXAMPLE 2 : Find the base 8 expansion of (12345) 10 Solution: First, divide l2345 by 8 to obtain 12345 = 8. 1543 + 1. 12345 = 8. 1543 + 1. Successively dividing quotients by 8 gives: 1543 = 8. 192 + 7, 1543 = 8. 192 + 7, 192 = 8 · 24 + 0, 192 = 8 · 24 + 0, 24 = 8 · 3 + 0, 24 = 8 · 3 + 0, 3 = 8 · 0 + 3. 3 = 8 · 0 + 3. Therefore, (12345) 10 = (3 0071 ) 8. 6/11/201633

35. Module #9 – Number Theory EXAMPLE 4 Find the hexadecimal expansion of (177130) 10 Solution: 177130 = 16 · 11070 + 10 177130 = 16 · 11070 + 10 11070 = 16 · 691 + 14, 11070 = 16 · 691 + 14, 691 = 16 · 43 + 3, 691 = 16 · 43 + 3, 43 = 16 · 2 + 11, 43 = 16 · 2 + 11, 2 = 16 · 0 + 2. 2 = 16 · 0 + 2.Therefore, (177 130) 10 = (2B3EA) 16 (177 130) 10 = (2B3EA) 16 6/11/201634

36. Module #9 – Number Theory 6/11/201635 Section 3.8: Matrices A matrix is a rectangular array of objects (usually numbers). An m  n matrix has exactly m rows, and n columns. m rows, and n columns. An n  n matrix is called a square matrix, whose order is n. 3  2  2 2  2

37. Module #9 – Number Theory 6/11/201636 Matrix Equality Two matrices A and B are equal iff they have the same number of rows, the same number of columns, and all corresponding elements are equal.Two matrices A and B are equal iff they have the same number of rows, the same number of columns, and all corresponding elements are equal.

38. Module #9 – Number Theory 6/11/201637 Row and Column Order The rows in a matrix are usually indexed 1 to m from top to bottom. The columns are usually indexed 1 to n from left to right. Elements are indexed by row, then column.The rows in a matrix are usually indexed 1 to m from top to bottom. The columns are usually indexed 1 to n from left to right. Elements are indexed by row, then column.

39. Module #9 – Number Theory 6/11/201638 Matrix Sums The sum A+B of two matrices A, B (which must have the same number of rows, and the same number of columns) is the matrix given by adding corresponding elements.The sum A+B of two matrices A, B (which must have the same number of rows, and the same number of columns) is the matrix given by adding corresponding elements. A+B = [a i,j +b i,j ]A+B = [a i,j +b i,j ]

40. Module #9 – Number Theory 6/11/201639 Matrix Products For an m  k matrix A and a k  n matrix B, the product AB is the m  n matrix: product AB is the m  n matrix: I.e., element (i, j ) of AB is given by the vector dot product of the i th row of A and the j th column of B (considered as vectors). (considered as vectors). Note: Matrix multiplication is not commutative!

41. Module #9 – Number Theory 6/11/201640 Matrix Product Example An example matrix multiplication to practice in class:An example matrix multiplication to practice in class: 2×3 ×4 3×4 ×4 2×4

42. Module #9 – Number Theory 6/11/201641 Identity Matrices The identity matrix of order n, I n, is the order-n matrix with 1’s along the upper-left to lower-right diagonal and 0’s everywhere else. A I n = AThe identity matrix of order n, I n, is the order-n matrix with 1’s along the upper-left to lower-right diagonal and 0’s everywhere else. A I n = A

43. Module #9 – Number Theory 6/11/201642 Powers of Matrices If A is an n  n square matrix and p  0, then: A p  AAA···A (A 0  I n )A p  AAA···A (A 0  I n ) Example:Example: p times

44. Module #9 – Number Theory 6/11/201643 If A is an m  n matrix, the transpose of A is the n  m matrix given by A tIf A is an m  n matrix, the transpose of A is the n  m matrix given by A t Matrix Transposition

45. Module #9 – Number Theory 6/11/201644 Symmetric Matrices A square matrix A is symmetric iff A=A t.A square matrix A is symmetric iff A=A t. Which is symmetric?Which is symmetric? A B C A B C

46. Module #9 – Number Theory 6/11/201645 Zero-One Matrices All elements of a zero-one matrix are 0 or 1 –Representing False & True respectively. The join of A, B (both m  n zero-one matrices):The join of A, B (both m  n zero-one matrices): A  B :  [a ij  b ij ] The meet of A, B:The meet of A, B: A  B :  [a ij  b ij ]

47. Module #9 – Number Theory 6/11/201646 Example A = B= A = B= We find the join between A  B = We find the meet between A  B =

48. Module #9 – Number Theory 6/11/201647 Boolean Products Let A be an m  k zero-one matrix, Let B be a k  n zero-one matrix,Let A be an m  k zero-one matrix, Let B be a k  n zero-one matrix, The Boolean product of A and B is like normal matrix product, But using  instead +The Boolean product of A and B is like normal matrix product, But using  instead + And using instead  And using instead  A⊙BA⊙B

49. Module #9 – Number Theory 6/11/201648 Boolean Powers For a square zero-one matrix A, and any k  0, the kth Boolean power of A is simply the Boolean product of k copies of A.For a square zero-one matrix A, and any k  0, the kth Boolean power of A is simply the Boolean product of k copies of A. A [k]  A ⊙ A ⊙ … ⊙ AA [k]  A ⊙ A ⊙ … ⊙ A k times

50. Module #9 – Number Theory 6/11/201649 Example A = B=A = B= A ⊙ B=

51. Module #9 – Number Theory 6/11/201650 ThenThen