The document discusses various topics related to working with numbers including:
1) Different types of numbers such as counting numbers, whole numbers, integers, rational numbers, and real numbers.
2) Divisibility tests for numbers 2-11 including using sums of digits or units places.
3) Methods for finding the highest common factor (HCF) and lowest common multiple (LCM) of numbers through division or using common divisors.
4) Examples of using divisibility tests and calculating HCF and LCM to solve word problems.
CBSE Class 10 Mathematics Real Numbers Topic
Real Numbers Topics discussed in this document:
Introduction
Rational numbers
Fundamental theorem of Arithmetic
Decimal representation of Rational numbers
Terminating decimal
Non-terminating repeating decimals
Irrational numbers
Surd
General form of a surd
Operations on surds
· Addition and subtraction
· Multiplication of surds
More Topics under Class 10th Real Numbers (CBSE):
Real numbers
Laws of
logarithms
Common and natural logarithms
Visit Edvie.com for more topics
CBSE Class 10 Mathematics Real Numbers Topic
Real Numbers Topics discussed in this document:
Introduction
Rational numbers
Fundamental theorem of Arithmetic
Decimal representation of Rational numbers
Terminating decimal
Non-terminating repeating decimals
Irrational numbers
Surd
General form of a surd
Operations on surds
· Addition and subtraction
· Multiplication of surds
More Topics under Class 10th Real Numbers (CBSE):
Real numbers
Laws of
logarithms
Common and natural logarithms
Visit Edvie.com for more topics
This is meant for age group 11 to 14 years.
For Class VIII CBSE.
Some viewers have requested me to send the file through mail.
So I allowed everybody to download.My request is whenever you are using plz acknowledge me.
Pratima Nayak ,Teacher,Kendriya Vidyalaya,Fort William,Kolkata
pnpratima@gmail.com
Based on Text book
Tips to prepare for Fundamentals of Quantitative Aptitude
Number Properties
LCM, HCF
Divisibility
Fractions & Decimals,
square
Square Roots
cyclicity
with shortcut tricks
This is meant for age group 11 to 14 years.
For Class VIII CBSE.
Some viewers have requested me to send the file through mail.
So I allowed everybody to download.My request is whenever you are using plz acknowledge me.
Pratima Nayak ,Teacher,Kendriya Vidyalaya,Fort William,Kolkata
pnpratima@gmail.com
Based on Text book
Tips to prepare for Fundamentals of Quantitative Aptitude
Number Properties
LCM, HCF
Divisibility
Fractions & Decimals,
square
Square Roots
cyclicity
with shortcut tricks
1 To 20 Divisibility Rules in Mathematics.pdfChloe Cheney
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He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
3. Introduction to Numbers
Type of Number Description
Counting Numbers {1, 2, 3, ...}
Whole Numbers {0, 1, 2, 3, ...}
Integers {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers p/q : p and q are integers, q is not zero
Irrational Numbers Not Rational
Real Numbers Rational Numbers and Irrational Numbers
Summary of Numbers
5. Divisibility Test
DIVISIBILITY BY 2
A number is divisible by 2, if its units digit is an even digit i.e. 0, 2, 4, 6, or 8.
DIVISIBILITY BY 3
A number is divisible by 3, if the sum of its digits is divisible by 3.
DIVISIBILITY BY 4
A natural number is divisible by 4, if the number formed by its digits in units and tens places is
divisible by 4.
Also, if the number formed by its digits in units and tens places is not divisible by 4, then the
given number is not divisible by 4.
Consider the number 79812. Since 12 is a multiple of 4. So, 79812 is divisible by 4.
Consider now the number n = 23472392. Since, 92 is a multiple of 4. So, n is divisible by 4. The
number m = 23798531 is not divisible by 4, because 31 is not a multiple of 4.
DIVISIBILITY BY 5
A number is divisible by 5 , if its unit digit is 0 or 5.
DIVISIBILITY BY 6
A number is divisible by 6 , if it is divisible by 2 as well as 3.
6. Divisibility Test
DIVISIBILITY BY 8
A natural number is divisible by 8, if the number formed by its digits in units, tens and hundreds
places is divisible by 8.
Consider the number 79816. Since 816 is a multiple of 8. So, 79816 is divisible by 8.
DIVISIBILITY BY 9
If the sum of the digits of a natural number is divisible by 9, then the number is divisible by 9.
ILLUSTRATION: If 21y5 is a multiple of 9, where y is a digit , what is the value of y?
SOLUTION: Since 21y5 is a multiple of 9. Therefore, the sum of its digits is a multiple of 9.
i. e. 2 + 1 + y + 5 is a multiple of 9
⇒ y + 8 is a multiple of 9
⇒ y + 8 = 0, or 9, or 18, or 27, 36, ... ...(i)
But, y is a digit. So, y can take values 0,1,2, …, 9.
.⋅. y + 8 can take values 8, 9, 10, 11, ..., 17. ...(ii)
From (i) and (ii), we get
y + 8 = 9 ⇒ y = 9 − 8 = 1
Hence,y = 1.
DIVISIBILITY BY 11
A number is divisible by 11, if the difference of its digits in odd
Places and the sum of its digits in even places is either 0 or a multiple of 11.
Consider three‐digit number 264. For this number, we have
Sum of the digits in odd places−Sum of the digits in even places = (2 + 4) − 6 = 0
So, 264 is divisible by 11.
Ex: 61809
7. Divisibility Test
Example: If 31z5 is a multiple of 3, where z is a digit, what might be the value of z?
Example: Without actual division find the remainder when 379843 is divided by 3.
8. Divisibility Test
Example: If 21y8 is a multiple of 6, where y is a digit, what might be the value of y?
Example: If 21y8 is a multiple of 6, where y is a digit, what might be the value of y?
9. Divisibility Test
Example: If 31z is a multiple of 11, where z is a digit, what is the value of z?
Example: Given that the number 148101a095 is divisible by 11, where a is some digit,
what are the possible values of a?
10. Divisibility Test
Example: Given that the number 1735538a05 is divisible by 9, where ′a′ is a digit, what are
the possible values of a?
Example: Given that the number 7713a8 is divisible by 4, where a is a digit, what are the
possible values of a?
12. HCF and LCM
Finding HCF
Method: (Short cut method) – Most popular method
2 |12, 16 (Divide by common divisor 2)
2 | 6, 8(Divide by common divisor 2)
3, 4
HCF = 2 x 2 = 4
Method: Continued Division Method
Divide the bigger number by smaller number. Then the divisors are divided in succession
by the remainders got. This division should be carried out till we get the remainder zero.
The last divisor is the HCF of the given numbers.
15. HCF and LCM
Example: Find the HCF and LCM of 12, 20, 32.
Example: The HCF of 72 and 252 is 36. Find their LCM.
16. HCF and LCM
Example: The students of a class can be divided into groups of 6 or groups of 8 without
leaving out any student. What will be the minimum number of students in such a class?
Example: A merchant has 120 liters and 180 liters of two kinds of oil. He wants to sell the
oil by filling the two kinds of oil in tins of equal volumes. What is the greatest volume of
such a tin?