3.4/3.5 The Integers and Division/ Primes and Greatest Common Divisors

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 adivides 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: • If a | b and a | c, then a | (b + c). • If a | b then a | bc. • 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).

3. Prove that if and , then .

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: 2164 399

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) andc  d (mod m) then (a)a+c b+d (mod m), and (b) ac  bd (mod m)

17. Prove andthen .

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 xn = (axn-1 + c) mod m Example: m = 11, a = 5, c = 2, x0=3 Example: m = 231–1, a = 75, c = 0

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

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 = akbk+ak-1bk-1+…+a1b+a0 where k is a non-negative integer, and a0, a1, …, ak are non-negative integers < b, and ak  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 akak-1ak-2…a1a0

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 = [aij]

36. Matrix Arithmetic • Addition and Subtraction • Scalar product

37. Matrix Multiplication • If A = [aij] and B = [bij], where A is an m by n matrix andBis an n by p matrix, then their product AB is the m by p matrix C = [cij] 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 cij = 0 for k:=1 to n do cij = cij + aikbkj end { The matrix [cij] 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 = [aij], we can define A2=AA, A3=AA2, etc. • Example:

44. Example: Find a formula for .

45. Transpose Matrix • For an m by n matrix A = [aij], we can define the transposeAt 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 = [bij] is A’s transpose, then for all relevant values of i and j, bij = aji • Example:

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

47. Zero-One Matrices • A zero-one matrix is one in which all the entries are zeros or ones. • The join of two matrices and is the “pairwise ‘or’” of their entries • The meet of two matrices and is the “pairwise ‘and’” of their entries

48. Zero-One Matrix Multiplication • If A = [aij] and B = [bij], where A is an m by n zero-one matrix andBis an n by p zero-one matrix, then their boolean productAB is the m by p matrix C = [cij] whose entries are given by

49. Examples =

50. Zero-One Matrix Powers For a zero-one matrix, define Example: