1. 3.4/3.5 The Integers and Division/ Primes and Greatest Common Divisors Let each of a and b be integers. We say that a divides b, in symbols a | b, provided that there exists an integer m for which b=am. Other ways of saying the same thing: –a is a divisor of b –a is a factor of b –b is a multiple of a –a goes evenly into b

2. Theorem For all integers a, b, and c: 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. Corollary: If a | b and a | c, then for all integers m and n we have a | (mb+nc).

4. Primes A prime is ….

5. The Fundamental Theorem of Arithmetic Every positive integer is either a prime or can be expressed as a product of primes in a unique way A composite is defined to be a positive integer > 1 which is not a prime.

6. Divisibility by 3 and 9 Theorem: An integer is divisible by 3 if and only if the sum of the digits in its decimal representation is divisible by 3. Theorem: An integer is divisible by 9 if and only if the sum of the digits in its decimal representation is divisible by 9.

7. Divisibility by 7 Theorem: A number of the form 10x + y is divisible by 7 if and only if x – 2y is divisible by 7. Examples: 399 2164

8. Theorem If n is a composite, then n has a prime divisor less than or equal to Let us use this fact to prove that 197 is prime.

9. Performing Prime Factorizations Use the above theorem, applied iteratively Example: 980

10. Theorem There are infinitely many primes

11. The Sieve of Eratosthenes

12. The “Division Algorithm” Let a be an integer and d a positive integer. Then there exist unique integers q and r for which (i) a = dq + r, and (ii) 0 ≤ r < d Our symbolism for q is a div d (the quotient), and for r it is a mod d (the remainder).

13. Greatest Common Divisor and Least Common Multiple

14. Theorem: Let p be a prime appearing m times in the prime factorization of a and n times in the prime factorization of b. Then (a) p appears times in the prime factorization of gcd(a,b), and (b) p appears times in the prime factorization of lcm(a,b).

15. Modular Arithmetic Define, for integers a and b and positive integer m, a  b (mod m)  m | (b – a) Theorems: 1. a  b (mod m)  a mod m = b mod m 2. a  b (mod m) 

16. Theorem If a  b (mod m) and c  d (mod m) then (a) a+c  b+d (mod m), and (b) ac  bd (mod m)

18. General Principle for Modular Arithmetic When the answer to your computation is to be a “mod m” result, you may discard multiples of m freely as you compute! Note that the remainder mod 9 of any integer is the same as the remainder mod 9 of the sum of its digits. Example: –What is (23459  49823 + 297) mod 9?

19. Example Today is On what day of the week will today’s date fall… –Next year? –Ten years from now? When will today’s date next fall on a ?

20. Definition Two integers a and b are said to be relatively prime provided gcd(a,b) = 1

21. Theorem For two positive integers a and b, the product gcd(a,b) lcm(a,b) is equal to the product ab.

22. Does the mod n Function work well as a hashing function? KEYS: 1880 1890 1900 1910 Etc. n = 15

23. Linear Congruential Pseudo- Random Number Generators x n = (ax n-1 + c) mod m Example: m = 2 31 –1, a = 7 5, c = 0 Example: m = 11, a = 5, c = 2, x 0 =3

24. Theorem: If a and b are positive integers, then gcd(a,b) = gcd(a, b mod a) 3.6 Integers and Algorithms

25. The Euclidean Algorithm procedure gcd(a, b: positive integers) x := a y := b while y  0 begin r := x mod y x := y y := r end { The gcd of a and b is now stored in the variable x }

27. Theorem Let b  Z, b > 1. Then any positive integer n can be uniquely expressed as n = a k b k +a k-1 b k-1 +…+a 1 b+a 0 where k is a non-negative integer, and a 0, a 1, …, a k are non-negative integers < b, and a k  0. This is our authority for using the “base b” expansion of the positive integer n, where specific symbols (like the arabic digits) are assigned to the integers a with 0 ≤ a < b and we can write the number n as a k a k-1 a k-2 …a 1 a 0

28. Examples Binary Octal Decimal Hexadecimal

29. Converting from Decimal to Binary Example: 190

30. Conversions Continued Decimal to hexadecimal Decimal to octal

31. Conversions Continued Hexadecimal to Decimal Octal to Decimal

32. Conversions Continued Binary to and from Hexadecimal Binary to and from Octal

33. Conversions Continued Octal to and from Hexadecimal – Just use binary as a go-between

34. 3.8 – Matrices A matrix is a rectangular array of numbers Notation

35. Special Cases If m = 1 we have a row matrix If n = 1 we have a column matrix Shorthand notation: A = [a ij ]

36. Matrix Arithmetic Addition and Subtraction Scalar product

37. Matrix Multiplication If A = [a ij ] and B = [b ij ], where A is an m by n matrix and B is an n by p matrix, then their product AB is the m by p matrix C = [c ij ] whose entries are given by

38. Example of Matrix Multiplication

39. Algorithm for Matrix Multiplication procedure multiply(A: m by n matrix, B: n by p matrix) for i:=1 to m do for j:=1 to p do begin c ij = 0 for k:=1 to n do c ij = c ij + a ik b kj end { The matrix [c ij ] is the matrix product of A and B }

40. Matrix-Chain Multiplication What is the most efficient way to compute a three-way product ABC, where A is m by n, B is n by p, and C is p by q? Grouping as (AB)C, we get mnp + mpq multiplications Grouping as A(BC), we get npq + mnq multiplications Theoretically, the result is the same, so we should choose the order which gives the fewest multiplications. Example: 5 by 3 times 3 by 4 times 4 by 2

42. The Identity Matrix For any positive integer n, the n by n matrices under matrix multiplication have an identity. It is

43. Powers of a Square Matrix For an n by n matrix A = [a ij ], we can define A 2 =AA, A 3 =AA 2, etc. Example:

45. Transpose Matrix For an m by n matrix A = [a ij ], we can define the transpose A t of A to be the n by m matrix whose rows are the columns of A and whose columns are the rows of A. i.e. if B = [b ij ] is A’s transpose, then for all relevant values of i and j, b ij = a ji Example:

46. Symmetric Matrices A square matrix A is said to be symmetric if A = A t

47. Zero-One Matrices

48. Zero-One Matrix Multiplication

49. Examples

50. Zero-One Matrix Powers Example:

51. Inverses