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Prime
• An integer greater than one is called a prime
number if its only positive divisors (factors) are one
and itself.
• Examples:
 The first six primes are 2, 3, 5, 7, 11 and 13.
 The prime divisors of 10 are 2 and 5.
 The Fundamental Theorem of Arithmetic shows
that the primes are the building blocks of the
positive integers: every positive integer is a product
of prime numbers in one and only one way, except
for the order of the factors. (This is the key to their
importance: the prime factors of an integer
determines its properties.)
Prime
 Algorithm to test whether an integer N>1 is prime:
Step1: N = 2 ? If so, N is prime, If not, continue.
Step2: 2 | N ? If so, N is not a prime, otherwise cont.
Step3: Compute the largest integer K ≤ √N. Then
Step4: D | N?
where D is any odd number such that
1 < D ≤ K. If D | N, then N is not prime,
otherwise, N is prime.
Greatest Common Divisor (GCD)
• Given two numbers not prime to one another, find their
greatest common divisor.
• GCD(a, b) = p1 min(a1, b1) p2 min(a2,b2) …pk min(ak, bk)
where p1, p2, p3,…., pk are prime factors of either a or b.
and some of ai and bi may be zeros.
• Example:
630 = 21. 3 2.5 1.7 1
450 = 2 1. 3 2.5 2.7 0
GCD(630, 450) = 2min(1, 1). 3 min(2, 2) 5min(1, 2). 7min(1, 0).
= 2 1. 3 2. 51. 7 0
= 90
Least Common Multiple (LCM)
• LCM(a, b) = p1 max(a1, b1) p2 max(a2,b2) …pk max(ak, bk)
where p1, p2, p3,…., pk are prime factors of either a or b.
and some of ai and bi may be zeros.
Example:
630 = 21. 3 2.5 1.7 1
450 = 2 1. 3 2.5 2.7 0
LCM(630, 450) = 2max(1, 1). 3 max(2, 2). 5max(1, 2). 7max(1, 0).
= 2 1. 3 2. 5 2. 7 1
= 3150
Euclidean Algorithm
• The algorithm is based on the following two
observations:
• If b|a then gcd(a, b) = b. This is indeed so
because no number (b, in particular) may have
a divisor greater than the number itself (I am
talking here of non-negative integers.)
• If a = bt + r, for integers t and r, then gcd(a, b)
= gcd(b, r).
Euclidean Algorithm
• Indeed, every common divisor of a and b also
divides r. Thus gcd(a, b) divides r. But, of
course, gcd(a, b)|b. Therefore, gcd(a, b) is a
common divisor of b and r and hence gcd(a, b)
= gcd(b, r). The reverse is also true because
every divisor of b and r also divides a.
Euclidean Algorithm
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Example
Let a = 2322, b = 654.
2322 = 654*3 + 360 gcd(2322, 654) = gcd(654, 360)
654 = 360*1 + 294 gcd(654, 360) = gcd(360, 294)
360 = 294*1 + 66 gcd(360, 294) = gcd(294, 66)
294 = 66*4 + 30 gcd(294, 66) = gcd(66, 30)
66 = 30*2 + 6 gcd(66, 30) = gcd(30, 6)
30 = 6*5 gcd(30, 6) = 6
Therefore, gcd(2322,654) = 6.
Euclidean Algorithm
• The greatest common divisor of 190 and 34 is
computed as follows using the Euclidean Algorithm:
190 = 5 * 34 + 20
34 = 1 * 20 + 14
20 = 1 * 14 + 6
14 = 2 * 6 + 2
6 =3*2+0
Since it is the next-to-last number appearing on the
right-hand side of these equations,
the GCD of the two is 2.
Euclidean Algorithm
• The greatest common divisor of 878 and 82 is
computed as follows via the Euclidean Algorithm:
878 = 10 * 82 + 58
82 = 1 * 58 + 24
58 = 2 * 24 + 10
24 = 2 * 10 + 4
10 = 2 * 4 + 2
4=2*2+0
Since it is the next-to-last number appearing on the
right-hand side of these equations,
the GCD of the two is 2.
Matrices
• Consider two families A and B.
• Every month, the two families have expenses
such as: utilities, health, entertainment, food,
etc.
• Let us restrict ourselves to: food, utilities, and
health.
• How would one represent the data collected?
• Many ways are available but one of them has
an advantage of combining the data so that it
is easy to manipulate them.
Matrices
• We will write the data as
follows:
If we have no problem confusing the names and what
the expenses are, then we may write
This is what we call a Matrix.
Matrix: Addition
• Addition of two matrices: Add entries one by one.
For example, we have
• Multiplication of a Matrix by a Number: In order
to multiply a matrix by a number, you multiply every
entry by the given number.
Matrix: Multiplication
Matrix: Multiplication
Matrices
• The size of the matrix is
given by the number of rows
and the number of columns.
If the two numbers are the
same, we called such matrix
a square matrix.
• Consider the matrix: its
diagonal is given by a
and d.
• For the matrix
Matrices
Its diagonal consists of a, e, and k. In general, if A is a
square matrix of order n and if aij is the number in the
ith-row and jth-column, then the diagonal is given by
the numbers aii, for i=1,..,n.
Upper-triangular and lower-triangular
matrices
• The diagonal of a square matrix helps define two type
of matrices: upper-triangular and lower-triangular.
• The diagonal subdivides the matrix into two blocks:
one above the diagonal and the other one below it.
• If the lower-block consists of zeros, we call such a
matrix upper-triangular.
• If the upper-block consists of zeros, we call such a
matrix lower-triangular.
Matrices
• For example, the matrices
are upper-triangular, while the matrices
are lower-triangular.
Transpose of a Matrix
Now consider the two matrices
• The matrices A and B are triangular. But there is something
special about these two matrices.
• If you reflect the matrix A about the diagonal, you get the
matrix B. This operation is called the transpose operation.
• Let A be a n x m matrix defined by the numbers aij, then
the transpose of A, denoted AT is the m x n matrix defined
by the numbers bij where bij = aji.
Transpose of a Matrix
• For example, for the matrix
we have
Matrices
• Properties of the Transpose operation. If
X and Y are m x n matrices and Z is an n x k
matrix, then
• 1.
– (X+Y)T = XT + YT
• 2.
– (XZ)T = ZT XT
• 3.
– (XT)T = X
Symmetric matrix
• Symmetric matrix is a
matrix equal to its transpose.
So a symmetric matrix must
be a square matrix. For
example, the matrices
are symmetric matrices.
Matrices
• A diagonal matrix is a symmetric matrix with all of its
entries equal to zero except may be the ones on the
diagonal. So a diagonal matrix has at most n different
numbers. For example, the matrices
are diagonal matrices. Identity matrices are examples of
diagonal matrices. Diagonal matrices play a crucial role
in matrix theory.
Invertible Matrices
• Invertible matrices are very important in many areas of
science. For example, decrypting a coded message uses
invertible matrices.
• Definition. An n x n matrix A is called nonsingular or
invertible if and only ifthere exists an n x n matrix B such
that
where In is the identity matrix.
The matrix B is called the inverse matrix of A. Example: