Solving Number Theory Problems

Introduction of solving number theory problems
Number theory is the theory about numbers which is called "the queen of mathematics" by the legendary mathematician Carl Friedrich Gauss, number theory is one of the oldest and largest branches of pure mathematics. The number theory delves deep into the structure and nature of numbers, and explores the remarkable, often beautiful relationships among them.

The number theory has many different types of numbers:

Solving Natural numbers problemssolving Prime numbers problems
solving Integers problems
solving Algebraic numbers problems
solving imaginary numbers problems
solving transcendental numbers problems


Example problems for solving number theory


Example: 1

Find the values of consecutive numbers where the sum of the two numbers is 161.

Solution:

Let as assume the two consecutive numbers be x, x+1.

Where the sum is 161 so,

x + x + 1 = 161

2x + 1 = 161

2x = 160

x = 80

Therefore, x + 1 = 80 + 1 = 81

So the consecutive numbers is 80 and 81.

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Example: 2

Find all integers n such that n − 18 and n + 18 are both perfect Squares.

Solution:            Let as assume n −18 = a2 and n +18 = b2.

Then b2 −a2 = 36, so (b−a) (b+ a) = 22 .32.

Because b−a and b+a are of the same parity,

So, the following possibilities are:  b − a = 2, b + a = 18, yielding b = 10, a = 8,

And b − a = 6, b+a = 6, yielding a = 0, b = 6.

Hence the integers with this property are n = 46 and n = 18.


Example :3


Find the values of consecutive numbers where the product of the two numbers is 121.

Solution:-

Let as assume the two consecutive numbers be x, x +1.

Where the product is 121 so,

x * (x + 1) = 121

x^2 + x = 121

x^2 + x – 121 = 0

x^2 + 12x – 11x – 121 = 0

(x + 12) (x - 11) = 0

(x + 12) = 0 (or) (x - 11) = 0

x = -12 (or) x = 11

-12 is not possible to get 121

Therefore we take x = 11

So, x + 1 = 11 + 1 = 12

The consecutive numbers is 11 and 12.

Mathematics Tangrams

Introduction to mathematics tangrams:

Among puzzles Tangram is certainly the mainly outstanding of each one. Tangram originates through China. Not everything is standard regarding its inventor or else correctly while the puzzle is invented. Eliminate it is recognized to be particularly accepted in China as of on 1800. The initial existing Chinese reserve lying on tangrams is obtainable in 1813. Having problem with Angle Obtuse keep reading my upcoming posts, i will try to help you.


Mathematics tangrams:

Some say in mathematics tangrams are typically played on residence during women as well as kids. This existence, the tangram is a problem game to know how to be enjoyed through the whole relations. It does not need an excessive quantity of ability. It now requires persistence, instance with mind. Someone by a part of article also a position of scissors know how to include consider of tangram, although particularly respected tangram locate include be entire since of carefully engraved ivory, tortoise casing with mother-of-pearl.

The confront of mathematics tangram be to organize seven uncomplicated geometrical part recognized tans two huge triangles, two undersized triangles with a rectangle within all type of behavior to build shape to symbolize community, substance also you know how to consider of. Our commission is just to reconstruct the shape, with every the seven tans exclusive of be related. But, several qualified puzzle solvers declare to group typically overrate their capability the complexity of the problem on the establishment period of live.

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Analysis tangrams:

According to statement, people who utilize their left understanding further lean to center lying on logical thoughts with accurateness. Those who utilize the right brain further center lying on aesthetics, reaction as well as originality.

Mathematics tangram knows how to further a further entire rational with aesthetic understanding. Skilled illustration - spatial thinker might locate to resolve mathematics tangrams exercise their valid analysis capability. And systematic academic might locate it improve their ability in concert by form, color along with thoughts.

The tangram is an analysis puzzle consisting of seven plane form, identified tans that are place mutually toward form shapes. The purpose of the problem is to structure a definite form by all seven parts that might not be related.

Two Perfect Squares

Definition:

In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number is that its square root is again an integer. Having problem with Solid Geometry keep reading my upcoming posts, i will try to help you.


Examples


For example 1,

(x + 2)(x + 2)

You get:

x^2 + 4x + 4

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Therefore, the quadratic expression x^2 + 4x + 4 is a perfect square since it factors into two identical binomials which are (x + 2) and (x + 2).

Notice that (x + 2) (x + 2) can be written (x + 2)2. So:

x^2 + 4x + 4 = (x + 2)2

For example 2,

(x + 3)(x + 3)

You get:

x^2 + 6x + 9

Therefore, the quadratic expression x^2 + 6x + 9 is a perfect square since it factors into two identical binomials which are (x + 3) and (x + 3).

Notice that (x + 3) (x + 3) can be written (x + 3)2. So:

x^2 + 6x + 9 = (x + 3)2

For example 3,

(x + 4)(x + 4)

You get:

x^2 + 8x + 16

Therefore, the quadratic expression x^2 + 8x + 16 is a perfect square since it factors into two identical binomials which are (x + 4) and (x + 4).

Notice that (x + 4) (x + 4) can be written (x + 4)2. So:

x^2 + 8x + 16 = (x + 4)2

For example 4,

(x + 5)(x + 5)

You get:

x^2 + 10x + 25

Therefore, the quadratic expression x^2 + 10x + 25 is a perfect square since it factors into two identical binomials which are (x + 5) and (x + 5).

Notice that (x + 5) (x + 5) can be written (x + 5)2. So:

x^2 + 10x + 25 = (x + 5)2

For example 5,

(x + 6)(x + 6)

You get:

x^2 + 12x + 30

Therefore, the quadratic expression x^2 + 12x + 30 is a perfect square since it factors into two identical binomials which are (x +6) and (x + 6).

Notice that (x + 6) (x + 6) can be written (x + 6)2. So:

x^2 + 12x + 30 = (x + 6)2

For example 6,

(x -5)(x -5)

You get:

x^2 - 10x + 25

Therefore, the quadratic expression x^2 - 10x + 25 is a perfect square since it factors into two identical binomials which are (x - 5) and (x - 5).

Notice that (x - 5) (x - 5) can be written (x - 5)2. So:

x^2 - 10x + 25 = (x - 5)2

Discrete Mathematics Notes

Introduction to discrete mathematics notes:

Discrete Mathematics deals with several selected topics in Mathematics that are essential to the study of many Computer Science areas. Since it is very difficult to cover all the topics, only two topics, namely “Mathematical Logic” and “Groups” have been introduced. These notes will be very much helpful to the students in certain practical applications related to Computer Science. In this article we shall discuss about discrete mathematics notes.

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Discrete mathematics notes:


Logical statement or Proposition:

A statement or a proposition is a sentence which is either true or false but not both.

A sentence which is both true and false simultaneously is not a statement, rather it is a paradox.

Example 1:

(a) Consider the following sentences:

(i) The earth is a planet.

(ii) Rose is a flower.

Truth value of a statement:

The truth of a statement is called its truth value. If a statement is true, we say that its truth value is TRUE or T and if it is false, we say that its truth value is FALSE or F.

Simple statements:

A statement is said to be simple if it cannot be broken into two or more statements. All the statements in (a) and (b) of Example 1 are simple statements.

Compound statements:

If a statement is the combination of two or more simple statements, then it is said to be a compound statement.

Conjunction:

If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement ‘p and q’ is called the conjunction of p and q and is written in the symbolic form as ‘p ? q’.

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Discrete mathematics notes problems:

Example 1:

(i) Show that ((~ p) ? (~ q)) ? p is a tautology.

Solution:

(i) Truth table for ((~ p) ? (~ q)) ? p

p          q         ~ p      ~ q      (~ p) ? (~ q)     ((~ p) ? (~ q))? p

T          T          F            F                      F                      T

T          F          F           T                      T                      T

F          T          T          F                      T                      T

F          F          T          T                      T                      T

The last column contains only T. Therefore ((~ p) ? (~ q)) ? p is a tautology.

Example 2:

Show that ((~ q) ? p) ? q is a contradiction.

Truth table for ((~ q) ? p) ? q

p          q          ~ q      (~ q) ? p          ((~ q) ? p) ? q

T          T          F          F                                  F

T          F          T          T                                  F

F          T          F          F                                  F

F          F          T          F                                  F

The last column contains only F. ? ((~ q) ? p) ? q is a contradiction.

Discrete Mathematics Sample

Introduction discrete mathematics sample

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Having problem with Graph Ordered Pairs keep reading my upcoming posts, i will try to help you.

Source Wikipedia


Discrete mathematics sample explanations:


Here will study about the discrete mathematics problems.

The discrete mathematics contains the set of topics. These topics are cover in Tautologies and Logical Equivalence

Sentential Functions and Sets are in logic and sets. Relation and functions,   Equivalence Relations Equivalence Classes. Natural numbers, division and factorization.  Division  ,Factorization ,

Greatest Common Divisor. these are the some of the discrete mathematics

Here we will see some of the samples in the discrete mathematics

Example for sentence

1.The sentence \1 + 2 = 3 and 2 + 2 = 4" is true.

2.The sentence \3 + 3 = 6 and _ is rational" is false.

Example for relations and functions

Definition. Let A and B be sets. By a relative R on A and B, we mean a subset of the Cartesian

Product a x b.

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Discrete mathematics sample problems:


Here we will learn about the discrete mathematics sample problems

The natural numbers

Example1:

The set of natural numbers is usually given by

N = {1, 2, 3 …}

Division

Examples

Division we divide 24 by 4

Solution:

24/4 =6

We divide 24 by 4 we get answer is 6

Factorization

(x² +9 ) to factorize the given problem

Solution:

The general form of the given equation is

(x² +a²) =(x+a) (x+a)

So. We factorize given problem

(x² +9 ) =(x+3) (x+3)

Greatest Common Divisor

Example:

12,4,36

We find the greatest common divisor of the given problems?

Solution:

We form the given problem

We divide by 4 all the numbers

12=2x2x 3

4=2x2 x 1

36=2x2x 9

We get the greatest common divisor of the given problems

Final answer is 2

Learning Perfect Number

Any number which is a positive integer  is called a perfect number if the sum of the factors(or divisors) of that number is equal to that number itself. Obviously, the factors excludes that number itself. In other words, we can say that a perfect number is an integer whose sum of factors(or divisors) is double the number.


Example of a perfect number

6

Factors of 6 are 1,2,3,6

Sum of the factors of 6(excluding 6) = 1+2+3 = 6 (the number itself)

Hence 6 is a perfect number.

Similarly, 28( factors of 28 excluding itself are 1,2,4,7 and 14, 1+2+4+7+14=28) is a perfect number.

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General form of a perfect number


The general form of a perfect number is given as 2p-1 (2p - 1), where p is a prime number. It is important to note that not all numbers of that form are perfect number but all perfect numbers will be of that form.

for p =2, 2p-1 (2p - 1) = 21 (22  - 1) =2*3 = 6, a perfect number.

Discrete Mathematics Relation

Introduction to discrete mathematics relation:

The discrete mathematics deals with functions and their properties, we noted the important property that all functions must have, namely that if a function does map a value from its domain to its co-domain, it must map this value to only one value in the co-domain.

Writing in set notation, if a is some fixed value:

` |{f(x)|x=a}|=1`

However, when we consider the relation, we relax this constriction, and so a relation may map one value to more than one other value. Having problem with Systems of Equations Solver keep reading my upcoming posts, i will try to help you.


Properties of discrete mathematics relation:-


In the following properties of discrete mathematics relation:-

Reflexive
Symmetric
Transitive
Antisymmetric
Trichotomy
Reflexive

Relation of the equality, = is reflexive. Examine that for all numbers a = a.So "=" is reflexive.

Symmetric

Relation is symmetric the values a and b:  a R b implies b R a.These type of relation is symmetric.

Transitive

Relation is transitive of all values of a, b, c: a R b and b R c implies a R c.These type of relation is transitive.

Antisymmetric

A relation is antisymmetric for all values a and b: a R b and b R a implies that a=b.These type of relation is antisymmetric

Trichotomy

A relation satisfy the all values a and b it holds true that: xRy or yRx.The  two relation numbers a and b, it is true that whether a ≥ b or b ≥ a (both if a = b).These type of relation is trichotomy. Please express your views of this topic How to Find Volume of a Cone by commenting on blog.


Example problems for discrete mathematics relation:-


Problem 1:-

`Let A = {1, 2, 3, 4, 5} and R : A harrA :-= {(a, b) : a |b}. What barR and R^-1?`

Solution:-

`{(1, 1) , (1, 2) , (1, 3) , (1, 4) , (1, 5) , (2, 2) , (2, 4) , (3, 3) , (4, 4) , (5, 5)}`

`barR = {(2, 1) , (2, 3) , (2, 5) , (3, 1) , (3, 2) , (3, 4) , (3, 5) , (4, 1) , (4, 2) , (4, 3) , (4, 5) , (5, 1) , (5, 2) , (5, 3) , (5, 4)}`

`R^-1 = {(1, 1) , (2, 1) , (3, 1) , (4, 1) , (5, 1) , (2, 2) , (4, 2) , (3, 3) , (4, 4) , (5, 5)}`

Problem 2:-

For each of the following relations of  pair which satisfies the relation ,and another pair which doesn't say whether each relation is reflexive relation,symmetric relation,transitive relation or anti-symmetric relation form discrete mathematics
(a) The relation on {1,2,3,4,5} defined by {(a,b)| a-b is even}
(b) The relation on {1,2,3,4,5} defined by {(a,b)| a+b is even}
(c) The relation on P, the set of all people,defined by {(a,b) | a and b have a common ancestor}

Solution:-

(a) The pair (4,2) satisfies the relation,(2,1) doesn't includes other relation.
(b) (3,3) satifies the relation,(3,2) doesn't includes other relation.
(c) (Bart,Lisa) satisfies the relation,(Homer,Marge) doesn't includes other relation.