2. Module #9 – Number Theory 1/5/20161 3. Algorithms, The Integers and Matrices

3. Module #9 – Number Theory 1/5/20162 Section 3.4: The Integers and Division Division: Let a, b  Z with a  0. a |b  “a divides b”. We say that “a is a factor of b”, “a is a divisor of b”, and “b is a multiple of a”. a does not divide b is denoted by a | b. We can express a | b using the quantifier  c (b = ac), Domain = Z. 3   12  True, but 3  7  False.

4. Module #9 – Number Theory 1/5/20163 The Divides Relations For every a, b, c  Z, we have 1. If a | b and a | c, then a | (b + c). 2. If a | b, then a | bc. 3. If a | b and b | c, then a | c. Examples:   3  12 and 3  9  3  (12 + 9)  3  21 (21 ÷ 3 = 7)   2  6  2  (6 × 3)  2  18 (18 ÷ 2 = 9)   4  8 and 8  64  4  64 (64 ÷ 4 = 16)

5. Module #9 – Number Theory 1/5/20164 The Division Algorithm let a be an integer and d a positive integer, then there exist unique integers q and r such that: a = d × q + r, 0  r < d. d is called divisor and a is called dividend. q is the quotient and r is the remainder (must be positive integer). q = a div d, r = a mod d.

6. Module #9 – Number Theory 1/5/20165 Examples What are the quotient and remainder when 101 is divided by 11? 101 = 11 × 9 + 2, q = 101 div 11 = 9, r = 101 – 11 × 9 = 2 = 101 mod 9 What are the quotient and the remainder when −11 is divided by 3? −11 = 3 × (−4) + 1, q = −4, r = 1 Note: − 11  3 × (− 3) − 2 because r can't be negative.

7. Module #9 – Number Theory 1/5/20166 Examples Find a and b if : 2a + b = 46 mod 7 and a + 2b = 47 div 9. 46 = 6 × 7 + 4 and 47 = 5 × 9 + 2 46 mod 7 = 4 and 47 div 9 = 5 2a + b = 4 (1) a + 2b = 5 (2) (1) – 2×(2) → – 3b = – 6 b = 2 and a = 1.

8. Module #9 – Number Theory 1/5/20167 Modular Congruence Theorem: Let a, b  Z, m  Z +. We say that a is congruent to b modulo m written a  b (mod m), if and only if 1) a mod m = b mod m, or 2) m | (a  b ) i.e. (a  b) mod m = 0.

9. Module #9 – Number Theory 1/5/2016 Examples Is 17 congruent to 5 modulo 6? 17 mod 6 = 5 and5 mod 6 = 5, also 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 and 14 mod 6 = 2, also 6 | (24 − 14)  6 | 10, then, 24 ≡ 14 (mod 6).

10. Module #9 – Number Theory 1/5/20169 Useful Congruence Theorems Let a, b, c, d  Z, m  Z +. If a  b (mod m) and c  d (mod m), then 1) 1) a + c  b + d (mod m), and 2) 2) ac  bd (mod m) e.g. Let 7 ≡ 2 (mod 5) and 11 ≡ 1 (mod 5) then: (7 + 11) ≡ (2 + 1) (mod 5)  18 ≡ 3 (mod 5) and (7 × 11) ≡ (2 × 1) (mod 5)  77 ≡ 2 (mod 5).

11. Module #9 – Number Theory 1/5/201610 Cryptology Caesar’s encryption method can be represented by the function f (p) = (p + 3) mod 26, 0 ≤ p ≤ 25. The letter “A” is replaced by 0, “B” by 1, …, and “Z” by 25. e.g. What is the secret message produce from the message “GOOD MORNING ZERO”? First replace the letters in the message with numbers p : 6 14 14 3 12 14 17 13 8 13 6 25 4 17 14 Second replace p by f (p) f (p) : 9 17 17 6 15 17 20 16 11 16 9 2 7 20 17 Translate this back to letters produces the encrypted message “JRRG PRUQLQJ CHUR ” To recover the original message from a secrete message. The inverse function f -1 (p) = (p – 3) mod 26 can be used. This process is called decryption.

12. Module #9 – Number Theory 1/5/201611 Hashing Functions Suppose that a computer has only the 20 memory locations 0, 1, 2, …, 19. Use the hashing function h where h(x) = (x + 5) mod 20 to determine the memory locations in which 57, 32, and 98 are stored. h(57) = (57 +5) mod 20 = 62 mod 20 = 2 h(32) = 37 mod 20 = 17 h(98) = 103 mod 20 = 3

13. Module #9 – Number Theory 1/5/201612 Section 3.5: Primes and Greatest Common Divisors A 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, 17,... Non-prime integer greater than 1 are called composite, because they can be composed by multiplying two integers greater than 1.

14. Module #9 – Number Theory 1/5/201613 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) 100 = 2·2·5·5 = 2 2 5 2 13 = 13 1024 = 2·2·2·2·2·2·2·2·2·2 = 2 10

15. Module #9 – Number Theory 1/5/201614 Theorem If n is a composite integer, then n has prime divisor ≤ e.g. 49  prime numbers less than are 2, 3, 5, 7 16  prime numbers less than are 2, 3. An integer n is prime if it is not divisible by any prime ≤ e.g. 13 where = 3.6 so the prime numbers are 2, 3 but non of them divides 13 so 13 is prime.

16. Module #9 – Number Theory 1/5/201615 To find the prime factor of an integer n : 1. Find 2. List all primes ≤ 2, 3, 5, 7, …, up to 3. Find all prime factors that divides n. Prime Factorization Technique

17. Module #9 – Number Theory 1/5/201616 Example 1

18. Module #9 – Number Theory 1/5/201617 Example 2

19. Module #9 – Number Theory 1/5/2016 Example 3

20. Module #9 – Number Theory 1/5/201619 Greatest Common Divisors 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. To find gcd: 1. Find all positive common divisors of both a and b, then take the largest divisor: e.g 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, so gcd (24, 36) = 12

21. Module #9 – Number Theory 1/5/201620 2. Use prime factorization: Take the minimum power of common factors Example: Find gcd(24, 180) 24 = 2×2×2×3 = 2 3 ×3 180 = 2×2×3×3 = 2 2 ×3 2 ×5 gcd(24, 36) = 2 min(3,2) ×3 min(1,2) ×5 min(0,1) = 2 2 ×3 1 = 12 Ways to Find GCD

22. Module #9 – Number Theory 1/5/201621 Find gcd (360, 3500)? 360 = 2 3 × 3 2 × 5 3500 = 2 2 × 5 3 × 7  gcd (360, 3500) = 2 2 × 5 = 20 Find gcd(17, 22)? No common divisors so gcd(17, 22) = 1 so, the numbers 17 and 22 are called relatively prime. Examples

23. Module #9 – Number Theory 1/5/201622 Least Common Multiple lcm(a, b) of positive integers, a and b, is the smallest positive integer that is a multiple both of a and of b. Find lcm(6, 10)? Take the maximum power of all factors 6 = 2 × 3, 10 = 2 × 5 lcm(6, 10) = 2 max(1,1) ×3 max(1,0) ×5 max(0,1) = 2×3×5 = 30 Find lcm (24, 180)? 24 = 2 3 × 3 1, 180 = 2 2 × 3 2 × 5  lcm (24, 36) = 2 3 × 3 2 × 5 = 360.

24. Module #9 – Number Theory Section 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. 1/5/201623

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

26. Module #9 – Number Theory Computers usually use binary notation (with 2 as the base) when carrying out arithmetic, and octal (base 8) or hexadecimal (base 16) notation when expressing characters, such as letters or digits. In fact, we can use any positive integer greater than 1 as the base when expressing integers. 1/5/201625 Representations of Integers

27. Module #9 – Number Theory Binary Expansions (Binary to Decimal) What is the decimal expansion of the integer that has (101011111) 2 as its binary expansion? Solution: We have (l 01011111) 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 = 256 + 64 + 16 + 8 + 4 + 2 + 1 = 351. 1/5/201626

28. Module #9 – Number Theory Decimal system 0123456789101112131415 0123456789ABCDEF Hexadecimal system Hexadecimal Expansions

29. Module #9 – Number Theory 28 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. (3016) 7 = 3·7 3 + 0·7 2 + 1·7 + 6 = (356) 10. Any to Decimal

30. Module #9 – Number Theory 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 3 = 2 · 1 + 1 1 = 2 · 0 + 1 Therefore, (241) 10 = (11110001) 2 29 Base Conversion (Decimal to Any)

31. Module #9 – Number Theory Find the base 8 expansion of (12345) 10. Solution: First, divide l2345 by 8 to obtain 12345 = 8 · 1543 + 1. Successively dividing quotients by 8 gives: 1543 = 8 · 192 + 7, 192 = 8 · 24 + 0, 24 = 8 · 3 + 0, 3 = 8 · 0 + 3. Therefore, (12345) 10 = (30071) 8. 30 Decimal to Any

32. Module #9 – Number Theory Find the hexadecimal expansion of (177130) 10. Solution: 177130 = 16 · 11070 + 10 11070 = 16 · 691 + 14, 691 = 16 · 43 + 3, 43 = 16 · 2 + 11, 2 = 16 · 0 + 2. Therefore, (177130) 10 = (2B3EA) 16 31 Example

33. Module #9 – Number Theory 32 Binary  Octal  Hexadecimal

34. Module #9 – Number Theory 1/5/201633 Section 3.8: Matrices An m×n matrix is a rectangular array of mn objects (usually numbers) arranged in m horizontal rows and n vertical columns. An n  n matrix is called a square matrix, whose order is n. 3  2 2222

35. Module #9 – Number Theory 1/5/201634 Matrix Equality Two matrices A and B are equal if and only if they have the same number of rows, the same number of columns, and all corresponding elements are equal.

36. Module #9 – Number Theory 1/5/201635 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.

37. Module #9 – Number Theory 1/5/201636 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. A + B = [a ij + b ij ]

38. Module #9 – Number Theory 1/5/201637 Matrix Products For an m  k matrix A and a k  n matrix B, the product AB is the m  n matrix: i.e. The 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). Note: Matrix multiplication is not commutative!

39. Module #9 – Number Theory 1/5/201638 Matrix Product Example 2×33×42×4

40. Module #9 – Number Theory 1/5/201639 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 = A

41. Module #9 – Number Theory 1/5/201640 Powers of Matrices If A is an n  n square matrix and p  0, then: A p  AAA ··· A (A 0  I n ) p times

42. Module #9 – Number Theory 1/5/201641 If A is an m  n matrix, then the transpose of A is the n  m matrix A T given by interchanging the rows and the columns of A. Matrix Transposition

43. Module #9 – Number Theory 1/5/201642 Symmetric Matrices A square matrix A is symmetric if and only if A T = A. Which is symmetric? A B C

44. Module #9 – Number Theory 1/5/201643 Zero-One Matrices All elements of a zero-one matrix are 0 or 1, representing False & True respectively. The join of A and B (both m  n zero-one matrices) is A  B :  [a ij  b ij ]. The meet of A and B is: A  B  [a ij  b ij ].

45. Module #9 – Number Theory 1/5/201644 Example A = B = The join between A and B is A  B = The meet between A and B is A  B =

46. Module #9 – Number Theory 1/5/201645 Boolean Products Let A be an m  k zero-one matrix and B be a k  n zero-one matrix, The Boolean product A ⊙ B of A and B is like normal matrix product, but using  instead + and using  instead .

47. Module #9 – Number Theory 1/5/201646 Boolean Powers For a square zero-one matrix A, and any k  0, the k th Boolean power of A is simply the Boolean product of k copies of A. A [k]  A ⊙ A ⊙ … ⊙ A k times

48. Module #9 – Number Theory 1/5/201647 Example Let A = and B = A ⊙ B =