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Module #9 – Number Theory
Chapter 3
Algorithms, Integers and Matrices
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Module #9 – Number Theory
Section 3.4: The Integers and
Division
• Division
• Let a,bZ with a0.
• a|b  “a divides b” : “cZ: b=ac”
So: a is a factor or a divisor of b, and b is a
multiple of a.
• a doesn’t divide b is denoted by a | b.
– Example: 312  True, but 37  False.
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Module #9 – Number Theory
Facts: the Divides Relation
• a,b,c  Z:
1. (a|b  a|c)  a | (b + c)
2. a|b  a|bc
3. (a|b  b|c)  a|c
Examples:
1. 312  39  3 (12+9)  321 (21÷3=7)
2. 26  2(6×3)  218 (18÷2= 9)
3. 48  8 64  464 (64÷ 4 = 16 )
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Module #9 – Number Theory
Composite and Prime Numbers
• 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...
• Non-prime integers greater than 1 are called
composite, because they can be composed
by multiplying two integers greater than 1.
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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 nondecreasing size. (tree or division)
100 = 2·2·5·5= 2252
13 = 13
1024 = 2·2·2·2·2·2·2·2·2·2=210
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Module #9 – Number Theory
Theorem
• If n is composite integer, then n has prime divisor
≤
n
• Ex: 49  prime numbers less than 49 are 2,3,5,7
16  prime numbers less than 16 are 2,3
• An integer n is prime if it is not divisible by any
prime ≤ n
• Ex: 13 where 13= 3.6 so the prime numbers are
2,3 but non of them divides 13 so 13 is prime
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Module #9 – Number Theory
• To find the prime factor of an integer n:
1- find n
2- list all primes ≤ n
2, 3, 5, 7,…root of n
3- find all prime factors that divides n.
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Module #9 – Number Theory
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Module #9 – Number Theory
The Division “Algorithm”
• let a be an integer and d a positive integer,
then there are 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
(positive integer)
 q = a div d , r =a mod d
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Module #9 – Number Theory
Examples
• What are the quotient and remainder when
101 is divided by 11?
101 = 11 × 9 + 2
, q = 9, r = 2
• 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  0
(can't be negative)
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Module #9 – Number Theory
Modular Congruence
Let a, b  Z, mZ+.
Thm:
a is congruent to b modulo m
written “a  b (mod m)”, iff
1) a mod m = b mod m
2) m | ab
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i.e. (ab) mod m = 0.
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Module #9 – Number Theory
Examples
Is 17 congruent to 5 modulo 6 ?
17 mod 6 = 5 and 5 mod 6 = 5
and 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
Then, 24 ≡ 14 (mod 6) and 6 | (24-14)  6 | 10
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Module #9 – Number Theory
Useful Congruence Theorems
• Let a,b,c,dZ, mZ+. Then if
ab (mod m) and cd (mod m), then:
▪ a+c  b+d (mod m), and
▪ ac  bd (mod m)
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Module #9 – Number Theory
Ex. :
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)
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Module #9 – Number Theory
Greatest Common Divisor (GCD)
• 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.
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Module #9 – Number Theory
Ways to find GCD
1.find all positive common divisors of both
a and b, then take the largest divisor
Ex: 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
 gcd (24, 36) = 12
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Module #9 – Number Theory
2. use prime factorization:
Take the min power of common factors
Example
Find gcd(24, 36)
24 = 2×2×2×3 = 23×3
36 = 2×2×3×3 = 22×32
gcd(24,36) = 22×3=12
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Module #9 – Number Theory
Ex: find gcd (120, 500)?
120 = 23 × 3 × 5
500 = 22 × 53
 gcd (120, 500) = 22 × 5 = 20
Ex: find gcd (17, 22)?
No common divisors so gcd (17, 22) = 1 so,
the numbers 17 and 22 are called relatively
prime.
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Module #9 – Number Theory
Least Common Multiple
• lcm(a,b) of positive integers a, b, is the
smallest positive integer that is a multiple
both of a and of b.
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Module #9 – Number Theory
Find lcm(6,10) ? Take the max power of all
Factors
6 = 2×3
10 = 2×5
Lcm(6,10) = 2×3×5=30
Find Lcm (24, 36)?
24 = 23 × 31
36 = 22 × 32
 Lcm (24, 36) = 23 × 32 = 72
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Module #9 – Number Theory
Section 3.1:
Algorithms
• A finite set of instructions for performing a
computation or for solving a problem
Examples:
- Finding the largest integer in a finite
sequence of integers
- Locating an element in a finite set
- Sorting elements in a finite sequence
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Module #9 – Number Theory
Finding the maximum element
procedure max(a1,a2,……..,an: integers)
max:=a1
for i:=2 to n
if max<ai then max:=ai
{max is the largest element}
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Module #9 – Number Theory
Linear Search Algorithm
procedure linearsearch (x:integer, a1,a2,..,an:integers)
i:=1
While(i≤n and x ≠ ai)
i = i+1
If i ≤ n then location:= i
else location:= 0
{location is the subscript of the term that equals x or the
location is 0 if x is not found }
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Module #9 – Number Theory
Bubble Sort Algorithm
procedure bubblesort(a1,a2,…..,an)
for i:=1 to n-1
for j:=1 to n-i
if aj > aj+1 then interchange aj and aj+1
{a1,…..,an is in increasing order}
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Module #9 – Number Theory
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.
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Module #9 – Number Theory
There are many important algorithms
involving integers besides those used in
arithmetic,
We will describe an algorithm for finding
the base b expansion of a positive integer for
any base b.
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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.
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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.
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Module #9 – Number Theory
BINARY EXPANSIONS
What is the decimal expansion of the integer
that has (l 0101 1111)2 as its binary
expansion?
Solution: We have
(l 0101 111l)2 = 1·28 + 0·27 + 1·26 + 0·25 + 1·24
+ 1 ·2 3 + 1 · 2 2 + 1 ·2 1 + 1 ·2 0
= 351 .
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Module #9 – Number Theory
HEXADECIMAL EXPANSIONS
Sixteen is another base used in computer
science. The base 16 expansion of an integer
is called its hexadecimal expansion.
hexadecimal digits used are 0, 1 , 2, 3, 4, 5, 6,
7, 8, 9, A, B, C, D, E, and F, where the letters
A through F represent the digits
corresponding to the numbers 10 through 15
(in decimal notation).
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Module #9 – Number Theory
EXAMPLE 2
What is the decimal expansion of the
hexadecimal expansion of (2AE0B)16 ?
Solution: We have
(2AE0B)16 = 2.164 +10.163 +14.162 +0·16 +11
= ( 175627)10 .
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Module #9 – Number Theory
BASE CONVERSION
EXAMPLE 1: 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 = (1111 0001)2
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Module #9 – Number Theory
EXAMPLE 2: 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 = (3 0071 )8 .
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Module #9 – Number Theory
EXAMPLE 4
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,
(177 130) 10 = (2B3EA) 16
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Module #9 – Number Theory
Section 3.8: Matrices
A matrix is a rectangular array of objects
(usually numbers).
2 3
An m  n matrix has exactly 

5

1


m rows, and n columns.
7
An n  n matrix is called
a square matrix,
whose order is n.
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0 
2 1
3 1


32
2 2
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Module #9 – Number Theory
Matrix Equality
• Two matrices A and B are equal iff they
have the same number of rows, the same
number of columns, and all corresponding
elements are equal.
 3 2  3 2 0
  1 6     1 6 0

 

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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.
 a1,1 a1, 2
a
a
2
,
1
2, 2

A  [ai , j ] 
 


am,1 am, 2
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 a1,n 

 a2 , n 
  

 am , n 
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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 = [ai,j+bi,j]
9
2 6   9 3 11
0  8   11 3   11  5

 
 

1
2  2 0   3
2 
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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:
k

AB  C  [ci , j ]   ai ,b , j 
  1

I.e., 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!
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Module #9 – Number Theory
Matrix Product Example
• An example matrix multiplication to
practice in class:
0  1 1 0
0 1  1 
1 0  5  1

2 0 3   2 0  2 0  3  2 11 3 

 1 0



3
1

2×3
3×4
2×4
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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
1

1 if i  j  0
I n  


0 if i  j 

0
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0
1

0




0

0


1
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Module #9 – Number Theory
Powers of Matrices
If A is an nn square matrix and p0, then:
• Ap  AAA···A (A0  In)
p times
• Example:
3
 2 1  2
  1 0    1

 
2

 1
1  2 1  2 1




0  1 0  1 0
1  3
2


0  2  1
3
4



3

2


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Module #9 – Number Theory
Matrix Transposition
• If A is an mn matrix, the transpose of A is
the nm matrix given by At
3
2 1
 0  1  2


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t
2 0 


 1  1
3  2
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Module #9 – Number Theory
Symmetric Matrices
• A square matrix A is symmetric iff A=At.
• Which is symmetric?
A
B
C
1 1  2 1 3  3 0 1 



1 1  1
0

1
0
2

1
 


 
1 1  3  1 2  1 1  2
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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, B (both mn zero-one
matrices):
A  B : [aij  bij]
• The meet of A, B:
A  B : [aij  bij]
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Module #9 – Number Theory
Example
A = 1 0 1
0 1 0 


B= 0 1 0
1 1 0 


1 1 1 
We find the join between A  B = 

1 1 0
We find the meet between A  B = 0 0 0
0 1 0 


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Module #9 – Number Theory
Boolean Products
• Let A be an m  k zero-one matrix,
Let B be a k  n zero-one matrix,
• The Boolean product of A and B is like normal
matrix product, But using  instead +
And using

instead 
A⊙B
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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
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Module #9 – Number Theory
Example
• A = 1
0


1
0
1

0

B=
1 1 0
0 1 1


A⊙B=
(1  1)  (0  0) (1  1)  (0  1) (1  0)  (0  1) 
(0  1)  (1  0) (0  1)  (1  1) (0  0)  (1  1) 


(1  1)  (0  0) (1  1)  (0  1) (1  0)  (0  1) 
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Module #9 – Number Theory
• Then
1 1 0
0 1 1


1 1 0
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