ICON College of Technology and Management
Department of Computing
HND in Computing
Unit 11: Maths for Computing
Lecturer: … Dr Asim Irshad….
Submitted by: Andrei Florin Lamasanu
ID No: 18278
Session: Winter 2022
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ICON College of Technology and Management
Department of Computing
HND in Computing
Unit 11: Maths for Computing
Lecturer: … Dr Asim Irshad….
Submitted by: Andrei Florin Lamasanu
ID No: 18278
Session: Winter 2022
Table of contents
- Part I............................................................................................................................................................
- GCD..........................................................................................................................................................
- LCM..........................................................................................................................................................
- Modular arithmetic..................................................................................................................................
- Modular Arithmetic as Remainders.....................................................................................................
- Multiplicative Modular arithmetic...........................................................................................................
- Public cryptography.................................................................................................................................
- Benefits of public cryptography...........................................................................................................
- The importance of prime numbers in computing field................................................................................
- The process of RSA..................................................................................................................................
- Part II.........................................................................................................................................................
- Binominal Distribution...........................................................................................................................
- Normal Distribution...............................................................................................................................
- Part III........................................................................................................................................................ - Vector graphics are used for:............................................................................................................. - system................................................................................................................................................ The distinctions between the arithmetic Cartesian coordinate system and the computer coordinate - In computer graphics, why are homogeneous coordinates used?.....................................................
- Part IV........................................................................................................................................................
- Maxima and Minima..............................................................................................................................
- Reference List............................................................................................................................................
- Programs evidence....................................................................................................................................
LCM
LCM is an acronym for "Least Common Multiple." The lowest number that may be
divided by both numbers is called the least common multiple of 2 numbers. It can be computed
using two or more numbers or fractions.
There are several ways to calculate the LCM of two integers. The prime factorization of
each number is one of the easiest ways to discover the LCM of two numbers, and then the
product of the greatest powers of the common prime factors is the LCM of those numbers
(Contini, 2006).
Few resolved examples:
The LCM of (12,14) = 84
The LCM of (11,14) = 154
The LCM of (10,4) = 20
To show my understanding of LCM have developed a program in PyCharm(Figure 2)
shown below, also the program can be found in appendices area.
Figure 2: The LCM code
Modular arithmetic
Modular arithmetic is a technique of integer-based arithmetic that considers the
remainder. In modular arithmetic, when numbers reach a specified fixed quantity (also known as
the modulus), they "wrap around" to leave a residual. A 12-hour clock is a good example of how
modular arithmetic may be used. If it is 10:00 now, the clock will reflect 3:00 instead of 15:00 in
5 hours. With a modulus of 12, 3 is the remainder of 15.
When divided by NN, a number xbmod NxmodN is equal to asking for the remainder of
xx. If two integers aa and bb have the same remainder when divided by NN, they are said to be
congruent (or in the same equivalence class). In this situation, we call it an equiv bpmod N.ab
(modN) (Houston, 2009).
Modular Arithmetic as Remainders
A
B
= Q remainder R
A is the dividend
B is the divisor
Q is the quotient
R is the remainder
Multiplicative Modular arithmetic
Modular multiplication may be found in a variety of branches of mathematics and has a
wide range of applications, such as cryptography, computer science, and computer algebra.
Properties of multiplication in modular arithmetic :
- If n x m = t, then n (mod N) * m(mod N) = t (mod N)
- If n = m (mod N), then kn = km (mod N) for any integer k.
- If n = m (mod N) and t = s (mod N), then nt = ms (mod N)
As example, we have 8 = 1 (mod 7) and 16 = 2 (mod 7), we have:
(816) = (12) = 2 (mod 7).
Public cryptography
Asymmetric cryptography, sometimes known as public key cryptography, is a group of
algorithms-based cryptographic protocols. The employment of two separate keys, one private or
secret and the other public, is required for this cryptographic approach. To protect data from
The process of RSA
RSA is an encryption method that is used to send communications securely over the
internet. It is founded on the idea that multiplying large numbers is simple, while factoring huge
numbers is complex. It is simple to confirm that 31 and 37 multiply to 1147, however finding the
factors of 1147 is a considerably more difficult task.
The following illustration (Figure 3) highlights how asymmetric cryptography works:
Figure 3: RSA structure (ResearchGate, n.d.)
So this is how RSA encryption is working. I'll find 2 massive prime numbers, p and q,
each with 100 or maybe 200 digits. I'm going to keep those 2 numbers private and I'm going to
multiply them together to obtain N = pq. That N is essentially my public key. It's quite simple for
me to obtain N; all I have to do is multiply my two numbers. However, if you know N, finding p
and q is very difficult. To obtain them, you must factor N, which seems to be a challenging task
(The science of encryption: prime numbers and mod n arithmetic, n.d.).
The following are the characteristics of the RSA:
The Rsa encryption is a finite field exponentiation over integers, including prime
values, that is widely used.
This method employs suitably large numbers, making it difficult to solve.
There are two sorts of keys in this method: private and public.
Problem 1
Given
a
1
d = 50
n = 150
Using the formula for arithmetic series:
Sn = n/2 [ 2 a 1 + (n-1) d]
S 10 = 150/2[(2) * (100) + (150-1) * (50)]
S 150 = £573,
As a result, the cost of painting a 150 m2 area is £573,750.
Problem 2
Given 3 middle terms:
11th term = a+10d
12th term = a+11d
13th term = a+12d
(a + 10d) + (a + 11d) + (a + 12*d) =
3a+3d = 720
a * (a+11*d)=
a + 11d=240 - (1)
21st term = a+20*d
22nd term = a+21*d
23rd term = a+22*d
T
21
+ T
22
+ T
23
A + 20d + a + 21d + a+ 22*d = 1320
3a + 63*d = 1320
3(a + 21*d) = 440
a + 21*d = 440 - (2)
Solving (1) & (2)
a + 11*d = 240
a + 21*d =4 40
-10*d = -
a + ar + ar
2
a+ ar+ ar
2
- ar
3
- ar
4
- ar
5
- ar
3
- ar
4
- ar
5
- r
3
(a + ar + ar
2
- r
3
r
3
r
3
r =
3
❑
a
2
Therefore:
a = 2
As result the first term a= 2 and common ratio r =
Problem 5
For finite terms, use the Geometric Sum Formula:
S
n
a ( r
n
( r − 1 )
Where given:
a = 4
r =
n-?
S
n
= a ¿ ¿
35
4 ( 1 −0.9 r
n
n
n
n
n
n
Using a logbook: log 0.125>n x log (0.9)
n <
log(0.125)
lo g (0.9)
n < 19.
As a result, there are 19 words that add up to more than 35.
Part II
Problem 2.
a)
When two dice are thrown, what are the chances of receiving a total of 9?
P (sample) =
b)
Sample space = {(H,T,T) (T,H,T) (T,T,H) (H,H,T) (H,T,H) (T,H,H) (H,H,H)}
3
c)
There are four sets of thirteen cards in the deck, and the probability for each card may be
computed as follows:
Probability(p)=
C
❑
4
1
C
❑
3
1
C
❑
52
2
4 x 3
d)
i. Probability that all three balls are blue can be calculated as:
Where:
b = binomial probability
x = total number of “successes” (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials.
e)
Probability 0 boy = C 0
(P )
5
q
0
Probability 1 boy = C 1
( P )
4
q
1
Probability 2 boys =
C
2
Probability 3 boys = C 3
(P )
2
q
3
Probability 4 boys = C 4
( P )
1
q
4
Probability 5 boys = C 5
(P )
0
q
5
Bellow is shown the graph of probability of distribution (Figure 5).
0 1 2 3 4 5
0.31 0.
Probability
Probability
Figure 5: Graph of probability distribution
f)
a) Normal distribution (x<19.5)
μ=
z =
x − μ
= 0.25, answer.
From z table we get = 0. 5987
b)
n(20 < x < 22)
It's possible to write it as n(x < 22) - n(x > 20)
z =
z =
From z table
= 0.3413 , answer.
Normal Distribution
The normal distribution is probably the most well-known and commonly utilised of all
distributions. It is compatible with a wide range of human attributes, including height, weight,
speed, and so on. Many living objects in nature, such as trees, animals, and insects, have a wide
range of traits. Many business and industrial factors are likewise regularly distributed.
Karl Gauss (1777 – 1855), who realised that the mistakes of repeated measurements of
things are typically normally distributed, is often credited with the invention of the normal curve.
The normal distribution is often referred to as the Gaussian distribution (Normal distribution, no
date).
The following are the features of the normal distribution:
It's a never-ending distribution.
distribution can take any value, but it will be limited to a range of 0 to 6 feet. In our inquiry, this
constraint is imposed physically.
The normal distribution, on the other hand, is unconcerned with the range. Even if the
range is extended from – to +, we can still discover a smooth curve. The Normal Distribution
then offers the probability of the value lying in a certain range for a given experiment for these
random variables, which are referred to as Continuous Variables. Also, by just entering the mean
and standard deviation values into the normal distribution calculator, you may determine the
probability density function (Normal distribution, no date).
Figure 7: Normal Distribution Curve (Siderova, 2018)
The Standard Normal Distribution Tables (Figures 8, 9) show the likelihood that z, the
Standard Normal Variable, is smaller than any values. Areas or probabilities are the regions
limited by the normal curve and the horizontal scale, whereas z values (values in the left column
and top row) are points on the horizontal scale.
Figure 8: Table of Standard Normal Probabilities for Negative z-scores (OpenStax CNX,
no date)
Part III
Problem 3.
Equation values are given (5, 7)
And is parallel to the following equation:
5x-7y = 4
5x -7y + k = 0
Substituting the following values in the formula:
(5)(5) - (7)(7) + k = 0
25- 49 + k = 0
Therefore k = 24, the equation is: 5x-7y +24 = 0
Problem 3.
Given:
2x - 7y + 5 = 0
As per perpendicular => m1 * m2= -
m2 = -1/m
So if given line = 2x+7y+5 = 0
Then the required line 7x-2y+k = 0
Using the values provided: (-1, -3)
(7) * (-1) - (2) * (-3) + k = 0
-7 + 6 + k = 0
-1 + k = 0
-1 + k = 0
k = 1
So here's the equation: 7x-2y+1 = 0
Problem 3.
Given tw equations:
6x + 5y+8 = 0
5x – hy + 8 = 0
Solution
m1 of 6x+5y+8=0 is -
m1 x m2 = -
h
h = 6
i) if they are parallel then:
m1 = m2, hence
h
h= -
Problem 3.
A type of picture is a vector graphic. Lines, curves, polygons, and other mathematical
objects are represented graphically in vector pictures. These visuals are computer-generated and
use the x and y axes as a reference definition (Lutkevich, 2021).
Vector graphics are used for:
Scalability. Vector formats are appropriate for activities that demand scalable graphics,
such as scalable type and text. Logos for companies and brands, for example, come in a
variety of sizes and can be seen in the corner of a mobile application or on a billboard on
the side of the road. A vector graphic logo may be adjusted up or down in size without
sacrificing quality or resulting in a large file.
Animation. Animated images are usually created as vector files, which produce cleaner
and smoother visuals.
Designing using a computer (CAD). Because of its scalability and simplicity of
updating mathematical formulae, CAD applications routinely employ vector files in
manufacturing, engineering, and design.