Lowest Prime Number

Introduction :

In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The first twenty-five prime numbers are:

Prime Numbers:- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Condition: - If x is the prime number then the next factors of the number x is 1 and X.                                                                                                                                                                                                                                        (Source:  Wikipedia)

Lowest Prime Number:

In prime numbers two is the very lowest prime number. It contains only two factors that is one and itself only. It has no third integer.

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Example Problems for Lowest Prime Number:

Problem 1:

Find out the number 31 prime number or not?

Solution:

The given number 31 is a prime number. Because it has no more values other than 1 and 31. Hence it is considered as a prime number.

Problem 2:

Find out the number 67 is prime number or not?

Solution:

The given number 61 is considered as a prime number. Because it is divisible by 1 and 67 only.

Problem 3:

Find out the number 117 is prime number or not?

Solution:

The given number 117 is not a prime number. The number divisible by more than two factors. Those factors are 1, 3, 9, 13, 39 and 117. Hence it is called not a prime number.

Problem 4:

Find out the number 217 is prime number or not?

Solution:

The given number is 217. So let us find that number is prime number or not.

The factors for the number 217 are 1, 7, 31 and 217. So the number contains more than two factors. Hence we cannot say it is a prime number.

Problem 5:

Find out the number 319 is prime number or not?

Solution:

The given number is 319. So let us find the given number is prime number or not. The prime factors for the given numbers are 1, 11, 29, and 319. Here the number 319 has more than two factors. So 319 is not a prime number.

Practice problem for lowest prime number:

Problem 1:

Find the number 53 is prime number or not?

Solution:

The getting number 53 is considered as a prime number.

Problem 2:

Find the number 89 is prime number or not?

Solution:

The getting number 89 is considered as a prime number.

Problem 3:

Find the number 78 is prime number or not?

Solution:

The given number 78 is not a prime number

Problem 4:

Find the number 110 is prime number or not?

Solution:

The given number 110 is not a prime number.

Prime Consecutive Numbers

Introduction to prime consecutive numbers:

A prime number is an number that has only two factors and is divisible only by 1 and itself, and by no other positive integer.The first few prime numbers are 2, 3, 5, 7, 11,13, 17, 19, 23, 29, 31 and 37.

The consecutive number is nothing but adjacent numbers for example the consecutive numbers of 2 is 3, even consecutive number between 1 to 10 is 2,4,6,8,10.

The prime consecutive number are of rare as the prime number were not of adjacent values normally. But there are two prime numbers that is also consecutive. the two and three are the only prime consecutive numbers. Let us see more about  prime consecutive numbers.

Discussion on Prime Consecutive Numbers

More about prime numbers:

A prime number is a natural number larger than 1 which cannot be expressed as the product of two smaller natural numbers.

Prime numbers were first studied rigorously by the ancient Greeks. One of the first theorems they proved was

Euclid’s theorem (˜ 300 BCE) There are infinitely many prime numbers.

Arithmetic progressions of primes

As we mentioned earlier, we are still unable to detect several types of patterns in the primes. However, we have made recent progress on one type of pattern, namely an arithmetic progression a, a + r, . . . , a + (k - 1)r.

In particular, for any given k, the primes contain infinitely many arithmetic progressions of length k. This result builds upon a number of existing results; for instance, in 1939, van der Corput showed that the primes contained infinitely many arithmetic progressions a, a + r, a + 2r of length three.

2
2, 3
3, 5, 7
5, 11, 17, 23
5, 11, 17, 23, 29
7, 37, 67, 97, 127, 157
7, 157, 307, 457, 607, 757
. . . . . . .  .  . ..  .  .  .  .   .   .   . .  . .
I am planning to write more post on find the prime factorization of 36, multiplying and simplifying rational expressions. Keep checking my blog.

More Discussion on Prime Consecutive Numbers

Let us take an example of number of prime numbers from the 1 to 1000 and try to find the number of prime consecutive numbers that can occur in that sequence.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.

Here if we notice the above given prime numbers from 1 to 1000 numbers. The ones digit of the prime numbers are all odd numbers except the 2, also the 2 and 3 are the consecutive numbers and also prime numbers.

The 2 and 3 are the only prime consecutive numbers as all the other higher values are  with odd values, however these may satisfy the condition of prime consecutive odd numbers. the consecutive number in the list of prime numbers do not occur.

Significant Relationship

Introduction to significant relationship:

In mathematics, the word significant indicates significant figures or significant digits. There are some rules to identify the significant digits. In this section, we shall discuss the significant relationship. The following rules and solved example problems will help the students to prepare for significant relationship.

Significant Relationship :

The following points represent relationship between the uncertainty estimates and the significant digits.

Fining a number to 3 significant digits shows that the relative improbability or uncertainty in that number is less than 1%.

If we know a number to 6 significant digits, the relative improbability or uncertainty is less than 0.001 %.

Some definitions for significant relationship can be shown as follows:

Relative uncertainty = `max (|described Value - True Value|/|True Value|)`

Absolute uncertainty = `max (|described Value| - |True Value|)`

Rules for identifying significant figures:

Rule 1:

All non zero figures in any number are significant.

That is,   353.76 has five significant figures.

In addition, 76 has two significant figures.

Rule 2:

Zeroes between the other figures are also significant.

That is,   23009 has five significant figures.

In addition, 9.02 has two significant figures.

Rule 3:

Zeroes located at the last part of decimal values are also significant.

That is,   2.450 has four significant figures.

In addition, 7.0 has two significant figures.

Rule 4:

Zeroes located at the start of decimal values are not significant.

That is,   0.00093 has two significant figures.

In addition, 0067 has two significant figures.

Then 0.000008 has one significant figure.

Rule 5:

Zeroes located at the last part of the whole number are indefinite.

For example, 7800 has at least two significant figures.

However, it may also has more than two significant figures as shown as follows.

7.8 x 103    has two significant figures

7.8 x 103   has three significant figures

7.800 x 103 has four significant figures.

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Example Problem for Significant Relationship :

Identify the number of significant digits from the following.

56, 0.023, 7.2530, 7003, 2300

Solution:

56 – The number 56 is a non-zero number that has two significant digits.

0.023 – Zeroes placed before the other numbers are not significant digits and the number of significant digits in 0.023 is two.

7.2530– Zeroes placed after the other numbers but after a decimal point are significant digits and the number of significant digits in 7.2530 is five.

7003 – The number 7003 has four significant digits because the Zeroes placed between the digits are always significant.

2300:

The number 2300 may have at least two significant digits. It also has three and four significant digits.

That is:

2300 – 2.3 x 103 has two significant digits.

2300 – 2.30 x 103 has three significant digits.

2300 – 2.300 x 103 has four significant digits.

Set Theory Venn Diagrams

INTRODUCTION TO SET THEORY VENN DIAGRAMS:

Define Set:

A set is a collection of well defined objects and these things which constitute a set are called its ‘elements’ or ‘members’.

Example:Set of odd numbers from 1 to 10 = {1, 3, 5, 7, 9}

Define Set theory - Venn diagram:

The geometrical representation of different types of sets is called “Venn diagram”

Example:

Basic Operations on Sets:

The primary operation on a set is,

- Union of sets:

The union of two nonempty sets A and B is the set consisting of all elements belonging to A or B or to both of them.

Ex: A = {1, 2, 5, 6} and B= {3, 7, 8, 9}

A `uu` B= {1, 2, 3, 5, 6, 7, 8, 9}

- Intersection of sets:

Let P and Q are two sets. The set of elements present in both the sets A and B are known as the set intersection of P and Q.

Ex P = {1, 2, 4, 6, 8}, Q = {6, 8, 9, 10, 12}.

P  `nn` Q = {6, 8}.

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Set Theorey Notations Used in Venn Diagrams

Let A, B are two sets and the following are the notations that are used in Venn diagrams:

AUB – >A union B which means that everything that is in either of the sets

A n B -> A intersect B which means that only the things that are in both of the sets

A c    -> A complement which means that everything in the object that is outside of A

A- B -> A minus B or A complement B which means everything in A except for anything in its overlap with B

- (AUB) -> not (A union B) which means everything outside of A and B

- (AnB) -> not (A intersect B) which means everything outer of the overlap of A and B



Set Theory Venn Diagrams Example Problem:

Mary asked 100 coffee drinkers whether they like cream or sugar in their coffee. 16 drinkers using cream only and 20 using both and 35 drinkers like sugar only.   How many like cream, sugar, cream but not sugar, sugar but not cream, cream and sugar, cream or sugar? Apply set theory Venn diagrams to solve.

Solution:We can find the solution for these questions using set theory Venn diagrams like

We should add 20 + 16 to find the no. of drinkers like cream

And 16+35 gives the number of sugar liking drinkers

Sugar but not cream is 35 and cream but not sugar is 16. Both gives 20 and we should add all 16+35+20 to find how many drinkers like cream or sugar which is 71.

Operations and Laws of Set Theory Venn Diagrams:

Operations on Sets:

Union of Sets
Intersection of sets
Disjoint of sets
Difference of two sets
Symmetrical difference of two sets
Complement of  a set
Algebraic laws:

Idempotent Laws
Identity laws
Associative laws
De-Morgan's Laws
Distributive laws

Linear Inequalities

Introduction

Two algebraic expressions or two real numbers related by the symbol '<', '>', '<=' or '>=' form an linear inequality.

Examples of numerical inequalities: 3 < 5; 7 > 5 x < 5;

Examples of literal inequalities: y > 2; x <= 3, y >= 4.

3 > x > 5 (read as x is greater than or equal to 3 and less than 5) and 2 < y < 4 are the examples of double inequalities.

How to solve linear inequality:

Added or subtracted by the same number on both side of the inequality.

Multiplied or divided by the same number on both side of the inequality but if we divide or multiply by a negative number, we must reverse the inequality sign.

Worked Examples for Linear Inequalities :

Example 1:

Solve the linear inequality: 5x – 3 < 7

Solution:

5x – 3 < 7

Add 3 on both side of the equation

5x - 3 + 3 < 7 + 3

5x < 10

Divide by 5 on both side of the equation

5x / 5 < 10 / 5

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Example 2:
Solve the linear inequality: 7x + 3 < 5x + 9

Solution:

7x + 3 < 5x + 9

Subtract 3 to the both side of the equation

7x + 3 - 3 < 5x + 9 -3

7x < 5x + 6

Subtract 5x on both sides of the equation

7x - 5x < 5x + 6 - 5x

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2x / 2 < 6 / 2

x < 3

So, the solution is (-infinity, 3).

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Additional Linear Inequality Examples

Example 3: Solve the linear inequality: 5x – 3 < 3x +1

Solution:

5x – 3 < 3x +1

Add 3 to the both side of the equation

5x – 3+3 < 3x +1+3

5x < 3x + 4

Subtract 3x on both sides of the equation

5x - 3x < 4

2x < 4

Divide by 2 on both side of the equation

2x / 2 < 4 / 2

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Example 4: Solve the linear inequality: 37 – (3x + 5) > 9x – 8 (x – 3)

Solution:

37 – (3x + 5) > 9x – 8 (x – 3)

37- 3x - 5 > 9x - 8x + 24

-3x + 32 > 1x + 24

Subtract x on both sides of the equation

-3x+32- x > 1x + 24 - x

-4x +32 > 24

Subtract 32 on both sides of the equation

-4x + 32-32 > 24 - 32

-4x > -8

Divide by -4 on both sides of the equation and change the inequality sign

-4x / -4 < -8 / -4

x < 2 So, the solution is (-infinity, 2)

Reflex Measure of an Angle

Introduction to reflex measure of an angle.

Let x be the given angle. Then we can define three important angles as follows:

1. The complimentary angle of x is 90 – x

2. The supplementary angle of x is 180 – x.

3. The reflex angle of x is 360 – x.

The above difference of angles must be known clearly.

Example: if x = 400, Then the reflex angle of x is given by 360 – x = 360 - 40 = 3200.

The above diagram, clearly shows that, the reflex angle of any angle is the remaining value of 360 beside the given angle x.

Now let us see some examples to understand and to practice these concepts in the following article.

Example Problems on Reflex Measure of Angle.
Ex 1: What is the reflex angle of  800?

Solution: The reflex angle of 80 is 360 - 80 = 2800.

Ex 2: Find the reflex of A is the given diagram Here XY is a straight line.

Solution: In the given diagram, A+15 = 180.

Therefore A = 180 -15 = 1650.

Therefore the reflex angle is given by 360 - A = 360 – 165 = 1950.

Ex 3: Find the reflex angle of B in the given diagram. Here XY is a straight line. Angle POY is 900.

Solution:

Since POY = 90

POY = 35+B = 90 implies B = 90-35

Therefore B = 550.

Therefore the reflex angle of B is 360 – B = 360 -55

= 3050.

Ex 4: Find the reflex angle of P in the given diagram. Here XY is a straight line.

Solution: Since XY is a straight line, Angle XOY = 180

implies 25+P = 180

implies P =180-25 =1550

Therefore the reflex angle of P is 360 – P = 360 -155 = 2050.



Ex 5: The reflex angle of P is 2250 . Find P.

Solution: Given: The reflex angle of P is 225 implies 360 - P = 225.

implies –P = 225 - 360 = -1350.

Therefore P =1350.

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Practice Problems on Reflex Measure of Angle

1. Find the reflex angle of 300.

[Ans: Reflex angle of 300 is 3300]

2. If the reflex angle of X is 1500, find the angle X.

[Ans: X = 2100]

I hope these examples would have helped you to find and measure the reflex angle for the given angle.

Introduction to Practice Number Line

Introduction :                                                                                           

A number line is a line on which real numbers are placed according to their value. For example, the number 3.5 or 3 1/2 corresponds with the point on a number line that is halfway between the numbers 3 and 4. Each point on a number line corresponds to a real number, and each real number has a unique point on the Number line that corresponds only to that number. Number lines are valuable tools which are easily used to illustrate the mathematical concepts such as subtraction, addition and positive and negative numbers.          

The number line is usually represented as a horizontal line. An arrowhead on either end of the line is suggests that the line  continues indefinitely in the positive and negative real number's which are denoted by ‘R’. The real numbers may be irrational and rational numbers, integers, whole numbers or also natural numbers.      

Steps to Draw a Number Line:

In order to draw a number line,

Draw a straight line with 1 inch divisions.
Position the bottom edge of your ruler at the start of the line. Mark each 1inch measurement divisions on the line until you reach the end of the line. Draw an arrow head (i.e., "<" and ">") on the left and right ends of the line.
Now, draw a perpendicular line at the centre and label it "0” where both the lines meet.
The divisions on the right hand side of the horizontal line will be the positive divisions i.e., numbers are prefixed with “+” sign and written as +1   +2   +3   +4   +5   +6……….

·       The divisions on the left hand side of the horizontal line will be the negative divisions i.e., numbers are prefixed with “-” sign and written as -1   -2   -3   -4    -5   -6………..

The points on a number line are called as the coordinates and the zero point is called the origin and the numbers on the right side of the origin are the positive numbers and those on the left side of the origin are the negative numbers. Hence the number line is divided as two symmetric halves by origin.

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Examples for Practice Based on Number Line:

Below are the examples on how to perform addition and subtraction using Number line

Ex 1: +6  - +2

Sol: +6 - +2 = +6  - 2 = +4

Ex 2: +8  +  +2

Sol: +8  +  +2 = +10

similarly,

+7 - +10 =    +7 - 10 =    -3
+7 + -10 =    +7 - 10 =    -3
+7+ +10 =    +7 + 10 =    +17
-7 - -10 =    -7 + 10 =    +3