Module 9 Number Theory 3 Algorithms The Integers

Module #9 – Number Theory 3. Algorithms, The Integers and Matrices 10/19/2021 1

Module #9 – Number Theory Section 3. 4: The Integers and 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, 10/19/2021 but 3 7 False. 2

Module #9 – Number Theory 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. 10/19/2021 4

Module #9 – Number Theory 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. 10/19/2021 5

Module #9 – Number Theory Examples • Find a and b if : 2 a + b = 46 mod 7 and a + 2 b = 47 div 9. 46 = 6 × 7 + 4 and 47 = 5 × 9 + 2 46 mod 7 = 4 and 47 div 9 = 5 2 a + b = 4 (1) a + 2 b = 5 (2) (1) – 2×(2) → – 3 b = – 6 b = 2 and a = 1. 10/19/2021 6

Module #9 – Number Theory 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. 10/19/2021 7

Module #9 – Number Theory Examples Is 17 congruent to 5 modulo 6? 17 mod 6 = 5 and 5 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). 10/19/2021

Module #9 – Number Theory Useful Congruence Theorems Let a, b, c, d Z, m Z+. If a b (mod m) and c d (mod m), then 1) a + c b + d (mod m), and 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). 10/19/2021 9

Module #9 – Number Theory 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) : 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. 10/19/2021 10

Module #9 – Number Theory 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. • • • 10/19/2021 h(57) = (57 +5) mod 20 = 62 mod 20 = 2 h(32) = 37 mod 20 = 17 h(98) = 103 mod 20 = 3 11

Module #9 – Number Theory 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. 10/19/2021 12

Module #9 – Number Theory 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 = 2252 13 = 13 1024 = 2· 2· 2· 2 = 210 10/19/2021 13

Module #9 – Number Theory Theorem • If n is a composite integer, then n has prime divisor ≤ e. g. 49 prime numbers less than 16 prime numbers less than are 2, 3, 5, 7 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. 10/19/2021 14

Module #9 – Number Theory Prime Factorization Technique • 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. 10/19/2021 15

Module #9 – Number Theory Example 1 10/19/2021 16

Module #9 – Number Theory Example 2 10/19/2021 17

Module #9 – Number Theory Example 3 10/19/2021

Module #9 – Number Theory 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 10/19/2021 19

Module #9 – Number Theory Ways to Find GCD 2. Use prime factorization: Take the minimum power of common factors Example: Find gcd(24, 180) 24 = 2× 2× 2× 3 = 23× 3 180 = 2× 2× 3× 3 = 22× 32× 5 gcd(24, 36) = 2 min(3, 2)× 3 min(1, 2)× 5 min(0, 1) = 22× 31 = 12 10/19/2021 20

Module #9 – Number Theory Examples • Find gcd (360, 3500)? 360 = 23 × 32 × 5 3500 = 22 × 53 × 7 gcd (360, 3500) = 22 × 5 = 20 • Find gcd(17, 22)? No common divisors so gcd(17, 22) = 1 so, the numbers 17 and 22 are called relatively prime. 10/19/2021 21

Module #9 – Number Theory 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 = 23 × 31 , 180 = 22 × 32 × 5 lcm (24, 36) = 23 × 32 × 5 = 360. 10/19/2021 22

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. 10/19/2021 23

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· 102 + 6· 10 + 5. However, it is often convenient to use bases other than 10.

Module #9 – Number Theory Representations of Integers 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. 10/19/2021 25

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· 28 + 0· 27 + 1· 26 + 0· 25 + 1· 24 + 1· 23 + 1· 22 + 1· 21 + 1· 20 = 256 + 64 + 16 + 8 + 4 + 2 + 1 = 351. 10/19/2021 26

Module #9 – Number Theory Hexadecimal Expansions Decimal system 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 A Hexadecimal system B C D E F

Module #9 – Number Theory Any to Decimal • What is the decimal expansion of the hexadecimal expansion of (2 AE 0 B)16 ? Solution: We have (2 AE 0 B)16 = 2· 164 + 10· 163 + 14· 162 + 0· 16 + 11 = (175627)10. • (3016)7 = 3· 73 + 0· 72 + 1· 7 + 6 = (356)10. 28

Module #9 – Number Theory Base Conversion (Decimal to Any) 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

Module #9 – Number Theory Decimal to Any Find the base 8 expansion of (12345)10. Solution: First, divide l 2345 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

Module #9 – Number Theory Example 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 = (2 B 3 EA)16 31

Module #9 – Number Theory Binary Octal Hexadecimal 32

Module #9 – Number Theory 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 10/19/2021 2 2 33

Module #9 – Number Theory 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. 10/19/2021 34

Module #9 – Number Theory 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. 10/19/2021 35

Module #9 – Number Theory 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 = [aij + bij] 10/19/2021 36

Module #9 – Number Theory 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 ith row of A and the jth column of B (considered as vectors). Note: Matrix multiplication is not commutative! 10/19/2021 37

Module #9 – Number Theory Matrix Product Example 2× 3 10/19/2021 3× 4 2× 4 38

Module #9 – Number Theory Identity Matrices • The identity matrix of order n, In , is the order-n matrix with 1’s along the upper-left to lower-right diagonal and 0’s everywhere else. • A In = A 10/19/2021 39

Module #9 – Number Theory Powers of Matrices • If A is an n n square matrix and p 0, then: Ap AAA ··· A (A 0 In) p times 10/19/2021 40

Module #9 – Number Theory Matrix Transposition • If A is an m n matrix, then the transpose of A is the n m matrix AT given by interchanging the rows and the columns of A. 10/19/2021 41

Module #9 – Number Theory Symmetric Matrices • A square matrix A is symmetric if and only if AT = A. • Which is symmetric? A 10/19/2021 B C 42

Module #9 – Number Theory 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 : [aij bij]. • The meet of A and B is: A B [aij bij]. 10/19/2021 43

Module #9 – Number Theory Example A= B = • The join between A and B is A B = • The meet between A and B is A B = 10/19/2021 44

Module #9 – Number Theory 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 10/19/2021 . 45

Module #9 – Number Theory 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. A[k] A⊙A⊙…⊙A k times 10/19/2021 46

Module #9 – Number Theory Example Let A = and B = A⊙B = 10/19/2021 47