Class 6th

Th ere are so many situations where we have to use negative numbers. Negative Numbers are the numbers with the negative sign. These numbers are less than zero.

Example

We use negative numbers to represent temperature.

Where +10 shows 10° hotter than 0 and -10 shows 10° colder than 0.

Successor and Predecessor

If we move 1 to the right then it gives the successor of that number and if we move 1 to the left then it gives the predecessor of that number.

Number	Predecessor	Successor
2	1	3
-8	-9	-7
-3	-4	-2
4	3	5

Tag me with a sign

• In the case of accounting we use negative sign to represent the loss and positive to represent the profit.
• In the case of sea level, we use a negative sign to represent the height of the place below the sea level and positive sign to represent the place above the sea level.

Integers

The collection of whole numbers and negative numbers together is called the Integers.

All the positive numbers are positive integers and all the negative numbers are negative integers. Zero is neither a positive nor a negative integer.

Representation of Integers on Number Line

To represent the integers on a number line, first, we have to draw a line and mark a point zero on it.

Then mark all the positive integers on the right side with the same distance as 1, 2, 3… and the entire negative numbers on the left side as -1,-2,-3…

Example

To mark (-7) we have to move 7 points to the left of zero.

Ordering of Integers

From the above number line, we can see that as we go to the right side the numbers are getting larger and as we move to the left the numbers are getting smaller.

Hence, any number on the right side on the number line is greater than the number on its left.

Example

5 is to the right of 2 so 5>2.

4 is to the right of -2, so 4>-2.

-4 is to the left of -1, so -4 < -1.

Some facts about Integers

• Any positive integer is always greater than any negative integer.
• Zero is less than every positive integer.
• Zero is greater than every negative integer.
• Zero doesn’t come in any of the negative and positive integers.

Addition of Integers

1. Addition of Two Positive Integers

If you have to add two positive integers then simply add them as natural numbers.

(+6) + (+7) = 6 + 7 = 13

2. Addition of Two Negative Integers

If we have to add two negative integers then simply add them as natural numbers and then put a negative sign on the answer.

(-6) + (-7) = – (6+7) = -13

3. Addition of One Negative and One Positive Integer

If we have to add one negative and one positive integer then simply subtract the numbers and put the sign of the bigger integer. We will decide the bigger integer ignoring the sign of the integers.

• (-6) + (7) = 1 (bigger integer 7 is positive integer)
• (6) + (-7) = -1(bigger integer 7 is negative integer)

Addition of Integers on a Number Line

1. Addition of Two Positive Integers

Example

Add 3 and 4 on the number line.

Solution

To add 3 and 4, first, we move 3 steps to the right of zero then again move 4 steps to the right from point 3.

As we reached to the point 7, hence (+3) + (+4) = +7

This shows that the sum of two positive integers is always positive.

2. Addition of Two Negative Integers

Example

Add (-2) and (-5) using a number line.

Solution

To add (-2) and (-5), first we move 2 steps to the left of zero then again move 5 steps to the left of (-2).

As we reached to the point (-7), hence (-2) + (-5) = -7.

This shows that the sum of two negative integers is always negative.

3. Addition of One Negative and One Positive Integer

a. If a positive integer is greater than the negative integer

To add (+6) and (-2), first we have to move 6 steps to the right from zero then move 2 steps to the left of point 6.

As we reached to the point 4, hence (+6) + (-2) = +4

b. If a negative integer is greater than the positive integer

To add (-5) and (+4), first we have to move 5 steps to the left of zero then move 4 steps to the right from point (-5).

As we reached to the point -1, hence (-5) + (+4) = (-1)

4. Additive Inverse

If we add numbers like (-7) and 7 then we get the result as zero. So these are called the Additive inverse of each other.

If we add (-2) + (2), then first we move 2 steps to the left of zero then we move two steps to the right of (-2).so finally we reached to zero.

Hence, if we add the positive and negative of the same number then we get the zero.

Example

What is the additive inverse of 4 and (-8)?

Solution

The additive inverse of 4 is (-4).

The additive inverse of (-8) is 8.

Subtraction of Integers on Number Line

If we subtract an integer from another integer then we simply add the additive inverse of that integer.

(-3) – (-2) = (-3) + 2 = -1

(-3) – (+2) = (-3) + (-2) = -5

1. Subtraction of Two Positive Integers

Example

Subtract 2 from 5.

Solution

To subtract 2 from 5, first, we move 5 steps to the right from zero then move 2 steps back to the left.

As we reached to 3 hence, 5 – (+2) = 5 – 2 = 3

2. Subtraction of Two Negative Integers

Example

Subtract (-12) from (-8).

Solution

To subtract (-12) from (-8), first, we have to move 8 steps to the left of zero then move 12 steps to the right of (-8).

As we reached to 4, hence (-8) – (-12) = (-8) + (12) = 4

3. Subtraction of One Negative and One Positive Integer

a. To subtract a positive integer from any other integer.

Example

Subtract 3 from (-4)

Solution

To subtract (-4) from (3), first, we have to move 4 steps to the left of zero then move 3 steps more to the left.

As we reached to (-7), hence (-4) – (+3) = (-4) + (-3) = -7.

b. To subtract a negative integer from any other integer

Example

Subtract (-3) from (4)

Solution

To subtract (-3) from (4), first, we have to move 4 steps to the right of zero then move 3 steps more to the right.

As we reached to (7), hence (4) – (-3) = (4) + (+3) = +7

The re are so many shapes around us made up of lines and curves like line segments, angles, triangles, polygons and circles etc. These shapes are of different sizes and measures.

Measuring Line Segments

A line segment is a fixed part of the line, so it must have some length. We can compare any line segment on the basis of their length.

1. Comparison by Observation

We can tell which line segment is greater than other just by observing the two line segments but it is not sure.

Here we can clearly say that AB > CD but sometimes it is difficult to tell which one is greater.

2. Comparison by Tracing

In this method we have to trace one line on paper then put the traced line segment on the other line to check which one is greater.

But this is a difficult method because every time to measure the different size of line segments we have to make a separate line segment.

3. Comparison using Ruler and a Divider

We can use a ruler to measure the length of a line segment.

Put the zero mark at point A and then move toward l to measure the length of the line segment, but it may have some errors on the basis of the thickness of the ruler.

This could be made accurate by using a Divider.

1. Put the one end of the divider on point A and open it to put another end on point B.
2. Now pick up the divider without disturbing the opening and place it on the ruler so that one end lies on “0”.
3. Read the marking on the other end and we can compare the two line.

Angles – “Right” and “Straight”

We can understand the concept of right and straight angles by directions.

There are four directions-North, South, East and West.

When we move from North to East then it forms an angle of 90° which is called Right Angle.

When we move from North to South then it forms an angle of 180° which is called Straight Angle.

When we move four right angles in the same direction then we reach to the same position again i.e. if we make a clockwise turn from North to reach to North again then it forms an angle of 360° which is called a Complete Angle.This is called one revolution.

In a clock, there are two hands i.e. minute hand and hour hand, which moves clockwise in every minute. When the clock hand moves from one position to another then turns through an angle.

When a hand starts from 12 and reaches to 12 again then it is said to be completed a revolution.

Acute, Obtuse and Reflex Angles

There are so many other types of angles which are not right or straight angles.

Angles	Meaning	Image
Acute Angle	An angle less than the right angle is called Acute angle.
Obtuse Angle	An angle greater than a right angle and less than straight angle is called Obtuse angle.
Reflex Angle	Angle greater than the straight angle is called Reflex angle.

Measuring Angles

By observing an angle we can only get the type of angle but to compare it properly we need to measure it.

An angle is measured in the “degree”. One complete revolution is divided into 360 equal parts so each part is one degree. We write it as 360° and read as “three hundred sixty degrees”.

We can measure the angle using a ready to use device called Protractor.

It has a curved edge which is divided into 180 equal parts. It starts from 0° to 180° from right to left and vice versa.

To measure an angle using protractor-

• Place the protractor on the angle in such a way that the midpoint of protractor comes on the vertex B of the angle.
• Adjust it so that line BC comes on the straight line of the protractor.
• Read the scale which starts from 0° coinciding with the line BC.
• The point where the line AB comes on the protractor is the degree measure of the angle.

Hence, ∠ABC = 72°

Perpendicular Lines

If two lines intersect with each other and form an angle of 90° then they must be perpendicular to each other.

Here AB and MN are intersecting at point N and form a right angle. We will write it as

AB ⊥ MN or MN ⊥ AB

Reads as AB is perpendicular to MN or MN is perpendicular to AB.

Perpendicular Bisector

If a perpendicular divides another line into two equal parts then it is said to be a perpendicular bisector of that line.

Here, CD is the perpendicular bisector of AB as it divides AB into two equal parts i.e. AD = DB.

Classification of Triangles

Triangle is a polygon with three sides. It is the polygon with the least number of sides. Every triangle is of different size and shape. We classify them on the basis of their sides and angles.

1. Classification on the basis of sides

Triangle	Meaning	Image
Scalene	If all the sides are different then it is called scalene triangle.
Isosceles	If two sides are equal then it is called isosceles triangle.
Equilateral	If all the sides are equal then it is called equilateral triangle.

2. Classification on the basis of Angles

Triangle	Meaning	Image
Acute Angled Triangle	If all the angles are less than 90° then this is called the acute-angled triangle.
Right Angled Triangle	If one of the angles is 90°then it is called the right-angled triangle.
Obtuse-angled Ariangle	If one of the angles of the triangle is obtuse angle then it is called Obtuse angled triangle.

Quadrilaterals

A polygon with four sides is called Quadrilateral.

S.No.	Name	Properties	Image
1.	Rectangle	It has two pairs of equal opposite sides.Opposite sides are parallel.All the angles are the right angle.
2.	Square	All the four sides are equal.Opposite sides are parallel.All the angles are the right angle.
3.	Parallelogram	It has two pairs of parallel opposite sides.Square and rectangle are also parallelograms.
4.	Rhombus	All the four sides are equal.Opposite sides are parallel.Opposite angles are equal.Diagonals intersect each other at the centre and at 90°.
5.	Trapezium	One pair of opposite sides is parallel.

Polygons

Any closed figure made up of three or more line segments is called Polygon.

We can classify the polygons on the basis of their sides and vertices –

Number of sides	Name of Polygon	Figure
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
9	Nonagon
10	Decagon
n	n-gon

Three-dimensional Shapes

The solid shapes having three dimensions are called 3D shapes.

Some of the 3D shapes around us

Cone		ice-cream cone
Cube		Block
Cuboid		Match-box
Cylinder		Glass
Sphere		Ball
Pyramid		Rubrics in a pyramid shape

Faces, Edges and Vertices

All the flat surfaces of the solid shape are called the Faces of that figure.

The line segment where the two faces meet with each other is called Edge.

The point where the two edges meet with each other is called Vertex.

No. of Faces, Edges and Vertices in some common 3- D shapes

S.No.	3 – D shape	Figure	Faces	Edges	Vertices
1.	Cube		6	12	8
2.	Cuboid		6	12	8
3.	Cone		2	1	1
4.	Sphere		1	1	0
5.	Cylinder		3	2	0
6.	Square based Pyramid		6	8	5
7.	Triangular Prism		5	9	6
8.	Rectangular Prism		6	12	8

The term ‘Geometry’ is derived from the Greek word ‘Geometron’. This has 2 equivalents. ‘Geo’ means Earth and ‘metron’ means Measurement.

Points

It is a position or location on a plane surface, which are denoted by a single capital letter.

Line Segment

It is a part of a line with the finite length and 2 endpoints.

The points A and B are called the endpoints of the segment.

It is named as:

Line

It is made up of infinitely many points with infinite length and no endpoint.

It extends indefinitely in both directions.

Named as:

Or sometimes

Intersecting Lines

The two lines that share one common point are called Intersecting Lines.

This shared point is called the point of intersection.

Here, line l and m are intersecting at point C.

Real life example of intersecting lines:

Parallel Lines

Two or more lines that never intersect (Never cross each other) are called Parallel Lines.

Real life examples of parallel lines:

Ray

It is a part of a line with one starting point whereas extends endlessly in one direction.

Real life examples of the ray are:

Curves

Anything which is not straight is called a curve.

1. Simple Curve – A curve that does not cross itself.

2. Open Curve – Curve in which its endpoints do not meet.

3. Closed Curve – Curve that does not have an endpoint and is an enclosed figure.

A closed curve has 3 parts which are as follows

1. Interior of the curve

It refers to the inside/inner area of the curve.

The blue coloured area is the interior of the figure.

2. The exterior of the curve.

It refers to the outside / outer area of the curve.

The point marked A depicts the exterior of the curve.

3. The boundary of the curve

It refers to the dividing line thus it divides the interior and exterior of the curve.

The black line which is dividing the interior and exterior of the curve is the boundary.

The interior and boundary of the curve together are called the curves “region”.

Polygons

It is a 2d closed shape made of line segments / straight lines only.

• Sides –It refers to the line segments which form the polygon, as in the above figure AB, BC, CD, DA are its sides.
• Vertex – Point where 2 line segments meet, as in the above figure A, B, C and D are its vertices.
• Adjacent Sides – If any 2 sides share a common endpoint they are said to be adjacent to each other thus called adjacent sides, as in the above figure AB and BC, BC and CD, CD and DA, DA and AB are adjacent sides.
• Adjacent Vertices – It refers to the endpoints of the same side of the polygon. As in the above figure A and B, B and C, C and D, D and A are adjacent vertices.
• Diagonals – It refers to the joins of the vertices which are not adjacent to each other. As in the above figure, AC and BD are diagonals of the polygon.

Angles

A figure formed from 2 rays which share a common endpoint is called Angle.

The rays forming the angle are known as its arms or sides.

The common endpoint is known as its vertex.

An angle is also associated with 3 parts

1. Interior – It refers to the inside/inner area.

The green coloured area is the interior of the angle.

2. Angle/boundary – It refers to the arms of the angle.

The red point is on the arm of the angle.

3. Exterior – It refers to the outside / outer area.

The blue point depicts the exterior of the figure.

Naming an Angle

While naming an angle the letter depicting the vertex appears in the middle.

Example

The above angle can also be named as ∠CBA.

An angle can also be named just by its vertex.

Example

Triangle

It is a 3 sided polygon. It is also the polygon with the least number of the sides.

Vertices: A, B and C

Sides: AB, BC and CA

Angles: ∠A, ∠B and ∠C

• Here, the light blue coloured area is the interior of the angle.
• The black line is the boundary.
• Whereas, the dark blue area is the exterior of the angle.

Quadrilaterals

It is a 4 sided polygon

Vertices: A, B, C, D

Sides: AB, BC, CD, DA

Angle: ∠A, ∠B, ∠C, ∠D

Opposite Sides: AB and DC, BC and AD

Opposite Angles: ∠B and ∠D, ∠A and ∠C

Adjacent Angles: ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, ∠D and ∠A.

Circles

It is a simple closed curve and is not considered as a polygon.

Parts of Circles

1. Radius – It is a straight line connecting the centre of the circle to the boundary of the same. Radii is the plural of ‘radius’.

2. Diameter –It is a straight line from one side of the circle to the other side passing through the centre.

3. Circumference – It refers to the boundary of the circle.

4. Chord – Any line that connects two points on the boundary of the circle is called Chord. Diameter is the longest chord.

5. Arc – It is the portion of the boundary of the circle.

6. Interior of the Circle – Area inside the boundary of the circle is called the Interior of the Circle.

7. The Exterior of the Circle – Area outside the boundary of the circle is called the Exterior of the Circle.

8. Sector– It is the region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.

9. Segment – It is the region in the interior of the circle enclosed by an arc and a chord.

Semi-circle

A diameter divides the circle into two semi-circles. Hence the semicircle is the half of the circle, which has the diameter as the part of the boundary of the semicircle.

The numbers which exactly divides the given number are called the Factors of that number.

As we can see that we get the number 12 by

1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2 and 12 ×1

Hence,

1, 2, 3, 4, 6 and 12 are the factors of 12.

The factors are always less than or equal to the given number.

Multiples

If we say that 4 and 5 are the factors of 20 then 20 is the multiple of 4 and 5 both.

List the multiples of 3

Multiples are always more than or equal to the given number.

Some facts about Factors and Multiples

• 1 is the only number which is the factor of every number.
• Every number is the factor of itself.
• All the factors of any number are the exact divisor of that number.
• All the factors are less than or equal to the given number.
• There are limited numbers of factors of any given number.
• All the multiples of any number are greater than or equal to the given number.
• There are unlimited multiples of any given numbers.
• Every number is a multiple of itself.

Perfect Number

If the sum of all the factors of any number is equal to the double of that number then that number is called a Perfect Number.

Perfect Number	Factors	Sum of all the factors
6	1, 2, 3, 6	12
28	1, 2, 4, 7, 14, 28	56
496	1, 2, 4, 8, 16, 31, 62, 124, 248, 496	992

Prime Numbers

The numbers whose only factors are 1 and the number itself are called the Prime Numbers.

Like 2, 3, 5, 7, 11 etc.

Composite Numbers

All the numbers with more than 2 factors are called composite numbers or you can say that the numbers which are not prime numbers are called Composite Numbers.

Like 4, 6, 8, 10, 12 etc.

Remark: 1 is neither a prime nor a composite number.

Sieve of Eratosthenes Method

This is the method to find all the prime numbers from 1 to 100.

Step 1: First of all cross 1, as it is neither prime nor composite.

Step 2: Now mark 2 and cross all the multiples of 2 except 2.

Step 3: Mark 3 and cross all the multiples of 3 except 3.

Step 4: 4 is already crossed so mark 5 and cross all the multiples of 5 except 5.

Step 5: Continue this process until all the numbers are marked square or crossed.

This shows that all the covered numbers are prime numbers and all the crossed numbers are composite numbers except 1.

Even and Odd Numbers

All the multiples of 2 are even numbers. To check whether the number is even or not, we can check the number at one’s place. If the number at ones place is 0,2,4,6 and 8 then the number is even number.

The numbers which are not even are called Odd Numbers.

Remark: 2 is the smallest even prime number. All the prime numbers except 2 are odd numbers.

Tests for Divisibility of Numbers

1. Divisibility by 2:

If there are any of the even numbers i.e. 0, 2, 4, 6 and 8 at the end of the digit then it is divisible by 2.

Example

Check whether the numbers 63 and 240 are divisible by 2 or not.

Solution: 

1. The last digit of 63 is 3 i.e. odd number so 63 is not divisible by 2.

2. The last digit of 240 is 0 i.e. even number so 240 is divisible by 2.

2. Divisibility by 3:

A given number will only be divisible by 3 if the total of all the digits of that number is multiple of 3.

Example 

Check whether the numbers 623 and 2400 are divisible by 3 or not.

Solution:

1. The sum of the digits of 623 i.e. 6 + 2 + 3 = 11, which is not the multiple of 3 so 623 is not divisible by 3.

2. The sum of the digits of 2400 i.e. 2 + 4 + 0 + 0 = 6, which is the multiple of 3 so 2400 is divisible by 3.

3. Divisibility by 4:

We have to check whether the last two digits of the given number are divisible by 4 or not. If it is divisible by 4 then the whole number will be divisible by 4.

Example

Check the number 23436 and 2582 are divisible by 4 or not.

Solution:

1. The last two digits of 23436 are 36 which are divisible by 4, so 23436 are divisible by 4.

2. The last two digits of 2582 are 82 which are not divisible by 4 so 2582 is not divisible by 4.

4. Divisibility by 5:

Any given number will be divisible by 5 if the last digit of that number is ‘0′ or ‘5′.

Example

Check whether the numbers 2348 and 6300 are divisible by 5 or not.

Solution: 

1. The last digit of 2348 is 8 so it is not divisible by 5.

2. The last digit of 6300 is 0 so it is divisible by 5.

5. Divisibility by 6:

Any given number will be divisible by 6 if it is divisible by 2 and 3 both. So we should do the divisibility test of 2 and 3 with the number and if it is divisible by both then it is divisible by 6 also.

Example

Check the number 342341 and 63000 are divisible by 6 or not.

Solution: 

1. 342341 is not divisible by 2 as the digit at ones place is odd and is also not divisible by 3 as the sum of its digits i.e. 3 + 4 + 2 + 3 + 4 + 1 = 17 is also not divisible by 3.Hence 342341 is not divisible by 6.

2. 63000 is divisible by 2 as the digit at ones place is even and is also divisible by 3 as the sum of its digits i.e. 6 + 3 + 0 + 0 + 0 = 9 is divisible by 3.Hence 63000 is divisible by 6.

6. Divisibility by 7:

Any given number will be divisible by 7 if we double the last digit of the number and then subtract the result from the rest of the digits and check whether the remainder is divisible by 7 or not. If there is a large number of digits then we have to repeat the process until we get the number which could be checked for the divisibility of 7.

Example

Check the number 2030 is divisible by 7 or not.

Solution:

Given number is 2030

1. Double the last digit, 0 × 2 = 0

2. Subtract 0 from the remaining number 203 i.e. 203 – 0 = 203

3. Double the last digit, 3 × 2 = 6

4. Subtract 6 from the remaining number 20 i.e. 20 – 6 = 14

5. The remainder 14 is divisible by 7 hence the number 203 is divisible by 7.

7. Divisibility by 8:

We have to check whether the last three digits of the given number are divisible by 8 or not. If it is divisible by 8 then the whole number will be divisible by 8.

Example

Check whether the number 74640 is divisible by 8 or not.

Solution:

The last three digit of the number 74640 is 640.

As the number 640 is divisible by 8 hence the number 74640 is also divisible by 8.

8. Divisibility by 9:

Any given number will be divisible by 9 if the total of all the digits of that number is divisible by 9.

Example

Check whether the number 2320 and 6390 are divisible by 9 or not.

Solution:

1. The sum of the digits of 2320 is 2 + 3 + 2 + 0 = 7 which is not divisible by 9 so 2320 is not divisible by 9.

2. The sum of the digits of 6390 is 6 + 3 + 9 + 0 = 18 which is divisible by 9 so 6390 is divisible by 9.

9. Divisibility by 10:

Any given number will be divisible by 10 if the last digit of that number is zero.

Example

Check the number 123 and 2630 are divisible by 10 or not.

Solution: 

1. The ones place digit is 3 in 123 so it is not divisible by 10.

2. The ones place digit is 0 in 2630 so it is divisible by 10.

Common Factors and Common Multiples

Example: 1

What are the common factors of 25 and 55?

Solution:

Factors of 25 are 1, 5.

Factors of 55 are 1, 5, 11.

Common factors of 25 and 55 are 1 and 5.

Example: 2

Find the common multiples of 3 and 4.

Solution:

Common multiples of 3 and 4 are 0, 12, 24 and so on.

Co-prime Numbers

If 1 is the only common factor between two numbers then they are said to be Co-prime Numbers.

Example

Check whether 7 and 15 are co-prime numbers or not.

Solution:

Factors of 7 are 1 and 7.

Factors of 15 are 1, 3, 5 and 15.

The common factor of 7 and 15 is 1 only. Hence they are the co-prime numbers.

Some more Divisibility Rules

1. Let a and b are two given numbers. If a is divisible by b then it will be divisible by all the factors of b also.

If 24 is divisible by 12 then 24 will be divisible by all the factors of 12(i.e.2, 3, 4, 6) also.

2. Let a and b are two co-prime numbers. If c is divisible by a and b then c will be divisible by the product of a and b (ab) also.

If 24 is divisible by 2 and 3 which are the co-prime numbers then 24 will also be divisible by the product of 2 and 3 (2×3=6).

3. If a and b are divisible by c then a + b will also be divisible by c.

If 24 and 12 are divisible by 4 then 24 + 12 = 36 will also be divisible by 4.

4. If a and b are divisible by c then a-b will also be divisible by c.

If 24 and 12 are divisible by 4 then 24 -12 = 12 will also be divisible by 4.

Prime Factorisation

Prime Factorisation is the process of finding all the prime factors of a number.

There are two methods to find the prime factors of a number-

1. Prime factorisation using a factor tree

We can find the prime factors of 70 in two ways.

The prime factors of 70 are 2, 5 and 7 in both the cases.

2. Repeated Division  Method

Find the prime factorisation of 64 and 80.

The prime factorisation of 64 is 2 × 2 × 2 × 2 × 2 × 2.

The prime factorisation of 80 is 2 × 2 × 2 × 2 × 5.

Highest Common Factor (HCF)

The highest common factor (HCF) of two or more given numbers is the greatest of their common factors.

Its other name is (GCD) Greatest Common Divisor.

Method to find HCF

To find the HCF of given numbers, we have to find the prime factorisation of each number and then find the HCF.

Example

Find the HCF of 60 and 72.

Solution:

First, we have to find the prime factorisation of 60 and 72.

Then encircle the common factors.

HCF of 60 and 72 is 2 × 2 × 3 = 12.

Lowest Common Multiple (LCM)

The lowest common multiple of two or more given number is the smallest of their common multiples.

Methods to find LCM

1. Prime Factorisation Method

To find the LCM we have to find the prime factorisation of all the given numbers and then multiply all the prime factors which have occurred a maximum number of times.

Example

Find the LCM of 60 and 72.

Solution:

First, we have to find the prime factorisation of 60 and 72.

Then encircle the common factors.

To find the LCM, we will count the common factors one time and multiply them with the other remaining factors.

LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 = 360

2. Repeated Division Method

If we have to find the LCM of so many numbers then we use this method.

Example

Find the LCM of 105, 216 and 314.

Solution:

Use the repeated division method on all the numbers together and divide until we get 1 in the last row.

LCM of 105,216 and 314 is 2 × 2 × 2 × 3 × 3 × 3 × 5 × 7 × 157 = 1186920

Real life problems related to HCF and LCM

Example: 1

There are two containers having 240 litres and 1024 litres of petrol respectively. Calculate the maximum capacity of a container which can measure the petrol of both the containers when used an exact number of times.

Solution:

As we have to find the capacity of the container which is the exact divisor of the capacities of both the containers, i. e. maximum capacity, so we need to calculate the HCF.

The common factors of 240 and 1024 are 2 × 2 × 2 × 2. Thus, the HCF of 240 and 1024 is 16. Therefore, the maximum capacity of the required container is 16 litres.

Example: 2

What could be the least number which when we divide by 20, 25 and 30 leaves a remainder of 6 in every case?

Solution:

As we have to find the least number so we will calculate the LCM first.

LCM of 20, 25 and 30 is 2 × 2 × 3 × 5 × 5 = 300.

Here 300 is the least number which when divided by 20, 25 and 30 then they will leave remainder 0 in each case. But we have to find the least number which leaves remainder 6 in all cases. Hence, the required number is 6 more than 300.

The required least number = 300 + 6 = 306.

Whole Numbers

• On adding the predecessor of 1, i.e., 0 in the queue of natural numbers, we get the whole number.
• 0,1,2,3,4,5……. are whole numbers.
• All whole numbers are natural numbers but all natural numbers are not whole numbers.

The Number Line

• The whole numbers are shown on the number line as shown below

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• The number line shows that the number on the right side of the other number is the greater number.
• The number line shows that the number on the left side of the other number is the smaller number.

Adding on the Number Line: 

• Suppose a+b is to be found from the number line. Then mark a unit on the number line and move the b units towards the right of a. 
• For example: The addition of 2 and 3 

Move 3 units towards the right of 2, we will get 5 

Subtracting on the Number Line:

• Suppose a−b is to be found from the number line then mark a on the number line then move b unit towards the left of a 
• For example: The subtraction of 5 and 3 

Move 3 units towards the left of 5, we will get 2 

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Properties of the Whole Number

1. Closure Property

• The whole numbers are closed under addition means the sum of two whole numbers is always a whole number.

For example: 5 and 8 are whole numbers and their sum 13 is also a whole number. 

• The whole numbers are also closed under multiplication, which means the multiplication of two whole numbers is always a whole number.

For example: 5 and 8 are whole numbers and their multiplication 40 is also a whole number. 

2. Commutative Property

• Whole numbers are commutative under addition. It means that they can be added in any order and the result will be the same.

For example: 4+2=6 and 2+4=6. 

• Whole numbers are also commutative under multiplication. It means that they can be multiplied in any order and the result will be the same.

For example: 5×3=15 and 3×5=15.

3. Associative Property 

• Whole numbers are associative under addition means rearranging the whole number in parenthesis and then adding will not affect the answer. 

For example: 

(12+5)+6

=17+6 

=23   

And 

12+(5+6) 

=12+11 

=23

• Whole numbers are associative under multiplication means rearranging the whole number in parenthesis and then multiplying will not affect the answer. 

For example: 

(2×5)×3

=10×3 

=30   

And 

2×(5×3) 

=2×15 

=30 

4. Distributivity of Multiplication Over Addition

• When a whole number is multiplied by the sum of the whole number then the distributive property of multiplication over addition is used. 

For example: 

8×(5+2) 

=(8×5)+(8×2) 

=40+16 

=56 

5. Additive Identity

• If adding 0 to any whole number gives the whole number itself, then 0 is the additive identity. 

For example: 9+0=9 

6. Multiplicative Identity

• If multiplying 1 to any whole number gives the whole number itself, then 1 is the multiplicative identity. 

For example: 6×1=6 

Numbers

• Symbols used for counting and measuring the objects are called numbers.

• A group of digits, denoting a number, is called a numeral.

• Writing a number in words is called numeration.

• 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are called digits or figures.

• Counting Numbers are called natural numbers.

• Counting Numbers alongwith zero is called whole numbers.

• Successor: The number that comes just after a given number. Example: Successor of 7 = 7+1=8

• Predecessor: The number that comes just before a given number. Example: Predecessor of 7 = 7 -1 = 6

Comparing Numbers

• The number with more digits > number with less digits. For example, 215 > 81.

• If two numbers have the same number of digits, then compare the digits on the extreme left and decide. If the extreme left digits are the same, compare the next digits to the right, and so on… For example: 57405926> 57405921.

• Ascending order means arrangement from the smallest to the greatest.

• Descending order means arrangement from the greatest to the smallest.

• The smallest four digit number is 1000 (one thousand). It follows the largest three digit number 999. 

• The smallest five digit number is 10,000. It is ten thousand and follows the largest four digit number 9999.

• The smallest six digit number is 100,000. It is one lakh and follows the largest five digit number 99,999. This carries on for higher digit numbers in a similar manner.

Indian System of Numeration

• We use ones, tens, hundreds, thousands, lakhs and crores.

• Commas are used to mark thousands, lakhs, and crores. For example: 3, 32, 40, 781 – Three crore thirty two lakh forty thousand seven hundred eighty one.

• Face value of a digit : The face value of a digit remains as it is, whatever place it may be occupying in the place value chart.

• Place value of a digit : The place value of digit in a numeral depends upon the place it occupies in the place value chart.

• Place value of a digit in a number = Face value × Position value.

International System of Numeration

• We use ones, tens, hundreds, thousands and millions. To express numbers larger than a million, a billion is used. 1 billion = 1,000 million.

• Commas are used to mark thousands and millions. For example: 3, 32, 40, 781 – Thirty three million two hundred forty thousand seven hundred eighty-one.

Large Numbers in Practice

Length

• 1 kilometre = 1000 metre

• 1 metre = 10 decimetre = 100 centimetre = 1000 millimetre

Mass

• 1 kilogram = 1000 grams

• 1 gram = 10 decigram = 100 centigram = 1000 milligram.

Capacity

• 1 litre = 10 decilitre = 100 centilitre = 1000 millilitre

Estimation (Rounding Off)

Nearest 10

• If the digit in the units place is less than 5, then the units digit is replaced by 0.

• If the digit in units place is greater or equal to 5, then the unit place is replaced by zero and tens place is increased by 1.

Nearest 100

• If the digit in the tens place < 5, then the tens and units place are replaced by zero.

• If the digit in the tens place is equal to or > 5 then the tens and units place is replaced by zero and the hundreds place is increased by 1.

Nearest 1000

• If the digit in the hundreds place is < 5, then the hundreds, tens and units place is replaced by 0.

• If the digit in the hundreds place is equal to or > 5, then the hundreds, tens and units place is replaced by 0 and the thousands place is increased by 1.

Roman Numerals

• Roman numerals are one of the early systems of writing numerals.

I = 1

II = 2

III = 3

IV= 4

V = 5

X = 10

L = 50

C = 100

D = 500

M = 1000

Rules of Roman Numeral System

• If a symbol is repeated, its value is added as many times as it occurs.

• A symbol is not repeated more than three times. But the symbols V, L and D are never repeated.

• If a symbol of smaller value is written to the right of a symbol of greater value, its value gets added to the value of greater symbol.

• If a symbol of smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol.

• The symbols V, L and D are never written to the left of a symbol of greater value, i.e. V, L and D are never subtracted. The symbol I can be subtracted from V and X only. The symbol X can be subtracted from L, M and C only.