3.4/3.5 the integers and division/ primes and greatest common divisors
Post on 30-Dec-2015
22 Views
Preview:
DESCRIPTION
TRANSCRIPT
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
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).
Prove that if and , then .
Primes
• A prime is ….
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.
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.
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
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.
n
Performing Prime Factorizations
• Use the above theorem, applied iteratively• Example: 980
Theorem
There are infinitely many primes
The Sieve of Eratosthenes
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).
Greatest Common Divisor and Least Common Multiple
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).
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) kmbaZk such that
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)
Prove and then .
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?
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 ?
Definition
• Two integers a and b are said to be relatively prime provided gcd(a,b) = 1
Theorem
• For two positive integers a and b, the product gcd(a,b) lcm(a,b) is equal to the product ab.
Does the mod n Function work well as a hashing function?
KEYS:
1880189019001910Etc.
n = 15
Linear Congruential Pseudo-Random Number Generators
xn = (axn-1 + c) mod m
Example: m = 231–1, a = 75, c = 0
Example: m = 11, a = 5, c = 2, x0=3
Theorem:
If a and b are positive integers, then
gcd(a,b) = gcd(a, b mod a)
3.6 Integers and Algorithms
The Euclidean Algorithm
procedure gcd(a, b: positive integers)
x := a
y := b
while y 0begin
r := x mod y
x := y
y := r
end
{ The gcd of a and b is now stored in the variable x }
Theorem
Let bZ, 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
Examples
• Binary
• Octal
• Decimal
• Hexadecimal
Converting from Decimal to Binary
• Example: 190
Conversions Continued
• Decimal to hexadecimal
• Decimal to octal
Conversions Continued
• Hexadecimal to Decimal
• Octal to Decimal
Conversions Continued
• Binary to and from Hexadecimal
• Binary to and from Octal
Conversions Continued
• Octal to and from Hexadecimal – Just use binary as a go-between
3.8 – Matrices
• A matrix is a rectangular array of numbers
• Notation
mnmm
n
n
aaa
aaa
aaa
A
...
............
...
...
21
22221
11211
Special Cases
• If m = 1 we have a row matrix• If n = 1 we have a column matrix• Shorthand notation: A = [aij]
mnmm
n
n
aaa
aaa
aaa
A
...
............
...
...
21
22221
11211
Matrix Arithmetic
• Addition and Subtraction
• Scalar product
5 [1 22 7 ]=¿
Matrix Multiplication
• If A = [aij] and B = [bij], 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 = [cij] whose entries are given by
n
kkjikij bac
pjmi
1
},,...,2,1{},,...,2,1{
Example of Matrix Multiplication
[ 1 2 4−1 − 3 0 ] [7 4
5 − 13 6 ]
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 }
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
The Identity Matrix
• For any positive integer n, the n by n matrices under matrix multiplication have an identity. It is
1...000
...............
0...100
0...010
0...001
nI
Powers of a Square Matrix
• For an n by n matrix A = [aij], we can define A2=AA, A3=AA2, etc.
• Example:
𝐴=[1 10 1]
Example:
Find a formula for .
Transpose Matrix
• For an m by n matrix A = [aij], we can define the transpose At 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:
[1 2 22 0 0
74 ]
𝑡
Symmetric Matrices
• A square matrix A is said to be symmetric if A = At
[ 3 0 3 73 0 127
−11250
1716
− 150164
]
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
[0 1 01 1 10 0 1 ]∨[ 1 1 0
0 0 01 1 1]
Zero-One Matrix Multiplication
• If A = [aij] and B = [bij], where A is an m by n zero-one matrix and B is an n by p zero-one matrix, then their boolean product A B is the m by p matrix C = [cij] whose entries are given by
kjik
n
k
ij bac
pjmi
1
},,...,2,1{},,...,2,1{
Examples
=
Zero-One Matrix Powers
[1 11 0 ]=𝐴
𝐴[2 ]=¿
For a zero-one matrix, define
Example:
For matrices (i.e. square matrices), we say that matrix has an inverse provided . We call the inverse of and denote by Not all square matrices have inverses. If a matrix does have an inverse, then it is unique.
Inverses
𝐴=[𝑎 𝑏𝑐 𝑑]𝐴−1=¿
Example:
top related