Number System (Quick Revision & Practice Questions)

Duration: 1 hr 21 min

This video lesson is available to enrolled students.

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Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. CS | 01 | Digital Logic and Circuits and Discrete Mathematical Structures: Digital Systems & Boolean Basics, Number System, Number Representation, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Permutation and Combination / क्रमचय और संचय
  3. CS | 02 | Computer Organization and Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Multiprocessors
  4. CS | 03 | Data Structures and Algorithm: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Complexity Theory, Advanced Algorithms
  5. CS | 04 | Problem Solving through C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  6. CS | 05 | Object Oriented Techniques: Object Oriented Technology, Java
  7. CS | 06 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Windows OS, Unix OS
  8. CS | 07 | Database Management Systems: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization, Query Processing & Optimization, File Organization & Indexing
  9. CS | 08 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics
  10. CS | 09 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  11. CS | 10 | Internet Technology, Web Design and Web Technology: Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, Emerging Technologies, JavaScript-Basics, HTML Basics
  12. CS | 11 | System Analysis And Design: Fundamentals of Systems and Information Systems, SDLC and Development Approaches, Software Documentation and Software Requirements Specification (SRS), SDD and Quality Assurance, Security, Risk Management & Maintenance
  13. CS | 12 | Information Security and Cyber Laws: Cyber Threats and Malware Attacks, Role of Internet and Web Services, Security in Mobile and Wireless Computing, Electronic Data Interchange and E-Commerce, Physical Security and Cryptography Basics, Intrusion Detection Systems (IDS), Cyber Safety and Security
  14. CS | 13 | Computer Graphics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing
  15. GS Paper | 01 | Indian and World Geography / भारत एवं विश्व का भूगोल: भारत का भौतिक पर्यावरण / India: Physical Environment, Computer Hardware, Geography as a Discipline and Geographical Thought / भूगोल: एक विषय एवं भौगोलिक चिंतन, पृथ्वी: उत्पत्ति, आंतरिक संरचना एवं विवर्तनिकी / The Earth: Origin, Interior and Tectonics, Geomorphic Processes and Landforms / भू-आकृतिक प्रक्रियाएँ एवं स्थलरूप, Atmosphere and Climate / वायुमंडल एवं जलवायु, Hydrosphere: Oceans / जलमंडल: महासागर, जैवमंडल, मृदा एवं पर्यावरण / Biosphere, Soils and Environment, Natural Hazards and Disasters / प्राकृतिक संकट एवं आपदाएँ, Human Settlements / मानव बस्तियाँ, संसाधन, कृषि, खनिज एवं ऊर्जा / Resources, Agriculture, Minerals and Energy, Industries, Services, Transport and Trade / उद्योग, सेवाएँ, परिवहन एवं व्यापार, प्रादेशिक एवं राजनीतिक भूगोल / Regional and Political Geography, Maps, Remote Sensing and Geographical Techniques / मानचित्र, सुदूर संवेदन एवं भौगोलिक तकनीकें, Fruit Science and Orchard Management / फल विज्ञान एवं बाग प्रबंधन, Vegetables and Specialty Crops / सब्जी एवं विशिष्ट फसलें, Indian Mural Painting / भारतीय भित्ति चित्रण, Analysis of Financial Statements / वित्तीय विवरणों का विश्लेषण, Computer Fundamentals
  16. GS Paper | 02 | History / इतिहास: भारतीय राष्ट्रीय आंदोलन एवं स्वतंत्र भारत / Indian National Movement and Independent India, Education of Children with Disabilities / दिव्यांग बच्चों की शिक्षा, भक्तिकाल, अपठित बोध, वैदिकसाहित्यं धर्मशास्त्रं संस्कृतिश्च, Sources and Historiography / इतिहास के स्रोत एवं इतिहास-लेखन, Prehistoric Cultures and the Harappan Civilisation / प्रागैतिहासिक संस्कृतियाँ एवं हड़प्पा सभ्यता, Vedic Age and Religious Movements / वैदिक काल एवं धार्मिक आंदोलन, महाजनपद एवं मौर्य साम्राज्य / Mahajanapadas and the Mauryan Empire, Post-Mauryan and Gupta Age / मौर्योत्तर एवं गुप्त काल, Harsha and Early Medieval India / हर्ष एवं पूर्व मध्यकालीन भारत, Delhi Sultanate and the Deccan Kingdoms / दिल्ली सल्तनत एवं दक्कन के राज्य, भक्ति एवं सूफी आंदोलन / Bhakti and Sufi Movements, Mughal Empire and the Marathas / मुगल साम्राज्य एवं मराठा, European Arrival, British Expansion and 1857 / यूरोपीय आगमन, ब्रिटिश विस्तार एवं 1857, औपनिवेशिक शासन, अर्थव्यवस्था एवं समाज सुधार / Colonial Rule, Economy and Social Reform, Ancient, Medieval and Early Modern World / प्राचीन, मध्यकालीन एवं प्रारंभिक आधुनिक विश्व, Modern and Contemporary World / आधुनिक एवं समकालीन विश्व, Society, Community and Social Groups / समाज, समुदाय और सामाजिक समूह, Religion, Education, Politics and Economy / धर्म, शिक्षा, राजनीति और अर्थव्यवस्था, Social Change and Development / सामाजिक परिवर्तन और विकास, Fundamentals, Media and Techniques of Art / कला के मूल तत्व, माध्यम और तकनीक, Buddhist, Jain and Hindu Art / बौद्ध, जैन और हिन्दू कला, Indian Architecture, Temple Sculpture and Bronzes / भारतीय स्थापत्य, मंदिर मूर्तिकला और कांस्य, Indian Miniature Painting Schools / भारतीय लघुचित्र शैलियाँ, Folk Art and Crafts / लोक कला और हस्तशिल्प, World Art: Prehistory to Romanticism / विश्व कला: प्रागैतिहासिक से स्वच्छन्दतावाद तक, Modern Western Art / आधुनिक पाश्चात्य कला, Printmaking / छापा कला, Charvaka, Jaina and Buddhist Philosophy / चार्वाक, जैन एवं बौद्ध दर्शन, Samkhya-Yoga Philosophy / सांख्य-योग दर्शन, Venn Diagrams, Syllogism and Inequalities / वेन आरेख, न्याय वाक्य एवं असमानता, Non-Verbal Reasoning / अशाब्दिक तर्क, भारतीय संविधान / Indian Constitution
  17. GS Paper | 03 | Indian Economy & Social Development / भारतीय अर्थव्यवस्था एवं सामाजिक विकास: भारतीय अर्थव्यवस्था / Indian Economy, जनसंख्या, मानव विकास एवं सांस्कृतिक भूगोल / Population, Human Development and Cultural Geography, Consumer Behaviour and Demand / उपभोक्ता व्यवहार एवं माँग, National Income Accounting / राष्ट्रीय आय लेखांकन, मुद्रा एवं बैंकिंग / Money and Banking, Determination of Income and Employment / आय एवं रोज़गार का निर्धारण, Inflation and Monetary Policy / मुद्रास्फीति एवं मौद्रिक नीति, Public Finance and Government Budget / लोक वित्त एवं सरकारी बजट, Agronomy and Farming Systems / शस्य विज्ञान एवं कृषि प्रणालियाँ, Agrometeorology / कृषि मौसम विज्ञान, Irrigation and Agricultural Engineering / सिंचाई एवं कृषि अभियांत्रिकी, Business Finance, Banking and Financial Markets / व्यावसायिक वित्त, बैंकिंग एवं वित्तीय बाज़ार, Food, Nutrition & Components of Food / भोजन, पोषण और भोजन के घटक
  18. GS Paper | 04 | General Science / सामान्य विज्ञान: कोशिका: संरचना एवं कार्य / Cell: Structure and Function, ऊष्मागतिकी एवं साम्यावस्था / Thermodynamics and Equilibrium, Scientific Achievements and Research / वैज्ञानिक उपलब्धियाँ और अनुसंधान, द्रव्य एवं रसायन विज्ञान की मूल अवधारणाएँ / Matter and Basic Concepts of Chemistry, परमाणु संरचना एवं नाभिकीय रसायन / Structure of Atom and Nuclear Chemistry, Periodic Classification and Chemical Bonding / आवर्त वर्गीकरण एवं रासायनिक आबंधन, States of Matter and Solutions / द्रव्य की अवस्थाएँ एवं विलयन, Redox Reactions and Electrochemistry / अपचयोपचय अभिक्रियाएँ एवं वैद्युतरसायन, Chemical Kinetics and Surface Chemistry / रासायनिक बलगतिकी एवं पृष्ठ रसायन, धातुकर्म, s-ब्लॉक एवं p-ब्लॉक तत्व / Metallurgy, s-Block and p-Block Elements, d- and f-Block Elements and Coordination Compounds / d- एवं f-ब्लॉक तत्व तथा उपसहसंयोजन यौगिक, Organic Chemistry: Basic Principles and Isomerism / कार्बनिक रसायन: मूल सिद्धांत एवं समावयवता, हाइड्रोकार्बन, हैलोऐल्केन एवं हैलोऐरीन / Hydrocarbons, Haloalkanes and Haloarenes, Organic Compounds Containing Oxygen and Nitrogen / ऑक्सीजन एवं नाइट्रोजन युक्त कार्बनिक यौगिक, Biomolecules, Polymers and Everyday Chemistry / जैव-अणु, बहुलक एवं दैनिक जीवन में रसायन, Analytical Chemistry and Spectroscopy / विश्लेषणात्मक रसायन एवं स्पेक्ट्रमिकी, Measurement and Mathematical Methods / मापन और गणितीय विधियाँ, गति और गति के नियम / Motion and Laws of Motion, कार्य, ऊर्जा, शक्ति और घूर्णी गति / Work, Energy, Power and Rotational Motion, Gravitation / गुरुत्वाकर्षण, Mechanical Properties of Matter / पदार्थ के यांत्रिक गुण, Thermal and Statistical Physics / तापीय और सांख्यिकीय भौतिकी, Oscillations and Mechanical Waves / दोलन और यांत्रिक तरंगें, Electrostatics and Current Electricity / स्थिरवैद्युतिकी और धारा विद्युत, Magnetism, Induction and Alternating Current / चुम्बकत्व, प्रेरण और प्रत्यावर्ती धारा, Optics and Photonics / प्रकाशिकी और फोटोनिकी, Quantum Physics and Relativity / क्वांटम भौतिकी और आपेक्षिकता, Atoms, Nuclei and Astrophysics / परमाणु, नाभिक और खगोल भौतिकी, जीव जगत एवं जैविक वर्गीकरण / Living World and Biological Classification, Plant Physiology / पादप कार्यिकी, मानव कार्यिकी / Human Physiology, Molecular Basis of Inheritance / वंशागति का आणविक आधार, Evolution / विकास, Biology in Human Welfare / मानव कल्याण में जीव विज्ञान, Biotechnology / जैव प्रौद्योगिकी, पारिस्थितिकी एवं पर्यावरण / Ecology and Environment, Family, Marriage and Kinship / परिवार, विवाह और नातेदारी, Soil Science / मृदा विज्ञान, Plant Nutrition, Manures and Fertilizers / पादप पोषण, खाद एवं उर्वरक, Floriculture, Landscaping and Protected Cultivation / पुष्प विज्ञान, भूदृश्य एवं संरक्षित खेती, Plant Breeding and Seed Technology / पादप प्रजनन एवं बीज प्रौद्योगिकी, Plant Protection / पादप संरक्षण, Anatomy and Physiology / शरीर रचना एवं शरीर क्रिया विज्ञान, Kinesiology and Biomechanics / कायनेसियोलॉजी एवं बायोमैकेनिक्स, Sports Psychology, Growth and Sociology / खेल मनोविज्ञान, वृद्धि एवं समाजशास्त्र, Sports Training and Physical Fitness / खेल प्रशिक्षण एवं शारीरिक पुष्टि, Sports Medicine, Injuries and Rehabilitation / खेल चिकित्सा, चोटें एवं पुनर्वास, Test, Measurement and Evaluation / परीक्षण, मापन एवं मूल्यांकन, Food and Nutrients / भोजन और पोषक तत्त्व, Nutrition Processes and Meal Planning / पोषण प्रक्रियाएँ और आहार नियोजन, Food Science, Processing and Food Safety / खाद्य विज्ञान, प्रसंस्करण और सुरक्षा, Public Nutrition, Health and First Aid / जन पोषण, स्वास्थ्य और प्राथमिक उपचार, Textile Science and Indian Textiles / वस्त्र विज्ञान और भारतीय वस्त्र, Extension Education and Development Communication / प्रसार शिक्षा और विकास संचार, Solid State Physics and Electronics / ठोस अवस्था भौतिकी और इलेक्ट्रॉनिकी, Main-group and Periodic Chemistry / मुख्य समूह एवं आवर्त रसायन, Biochemistry and Bioenergetics / जैवरसायन एवं जैवऊर्जिकी, Non-chordate Diversity / अरज्जुकी विविधता, Plant Morphology and Anatomy / पादप आकारिकी और शारीर, Animal Reproduction and Development / जंतुओं में जनन एवं परिवर्धन
  19. GS Paper | 05 | Elementary Maths / प्रारंभिक गणित: Number System / संख्या पद्धति, Divisibility Rules / विभाज्यता के नियम, HCF and LCM / महत्तम समापवर्तक और लघुत्तम समापवर्त्य, Percentage / प्रतिशत, Profit, Loss and Discount / लाभ, हानि और बट्टा, Simple and Compound Interest / साधारण और चक्रवृद्धि ब्याज, Ratio and Proportion / अनुपात और समानुपात, Average / औसत, Time and Work / समय और कार्य, Time, Speed and Distance / समय, चाल और दूरी, Algebra / बीजगणित, Solid Mensuration / घन क्षेत्रमिति, Data Interpretation / आँकड़ा निर्वचन, Arithmetic / अंकगणित, Statistics / सांख्यिकी, Probability / प्रायिकता, अनुक्रम, श्रेणी और संयोजन / Sequences, Series and Combinatorics, Mixture and Alligation / मिश्रण और एलिगेशन, Calculus / कलन, Trigonometry / त्रिकोणमिति, Vectors / सदिश, Number Systems and Number Theory / संख्या पद्धति और संख्या सिद्धांत, Euclidean Geometry and Mensuration / यूक्लिडीय ज्यामिति और क्षेत्रमिति, Statistics and Probability / सांख्यिकी और प्रायिकता, Coordinate Geometry, Vectors and 3D Geometry / निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति, Polynomials, Equations and Complex Numbers / बहुपद, समीकरण और सम्मिश्र संख्याएँ, Plane Mensuration / समतल क्षेत्रमिति, Simplification, Indices and Surds / सरलीकरण, घातांक और करणी, Sets, Relations, Functions and Discrete Mathematics / समुच्चय, संबंध, फलन और विविक्त गणित, संज्ञानात्मक और रचनावादी अधिगम / Cognitive and Constructivist Learning, आव्यूह और रैखिक बीजगणित / Matrices and Linear Algebra, वास्तविक और सम्मिश्र विश्लेषण / Real and Complex Analysis, Mathematics Pedagogy / गणित शिक्षणशास्त्र, Production and Costs / उत्पादन एवं लागत, Statistics for Economics / अर्थशास्त्र में सांख्यिकी, Series / श्रेणी, Analogy and Classification / सादृश्यता एवं वर्गीकरण, Mathematical Operations and Missing Numbers / गणितीय संक्रियाएँ एवं लुप्त संख्या, Direction and Distance / दिशा एवं दूरी, Trigonometry / त्रिकोणमिति, Integral Calculus and Vector Analysis / समाकल कलन और सदिश विश्लेषण, अवकल समीकरण और अनुप्रयुक्त गणित / Differential Equations and Applied Mathematics, Age Problems / आयु संबंधी प्रश्न, Partnership / साझेदारी
  20. GS Paper | 06 | Indian Polity and Governance / भारतीय राजव्यवस्था एवं शासन: भारतीय संविधान / Indian Constitution, Political Science and the State / राजनीति विज्ञान और राज्य, Political Ideologies / राजनीतिक विचारधाराएँ, World Political Thought / विश्व राजनीतिक चिंतन, Indian Political Thought / भारतीय राजनीतिक चिंतन, Comparative Politics / तुलनात्मक राजनीति, संघ सरकार / Union Government, संघवाद, राज्य और स्थानीय शासन / Federalism, State and Local Government, भारतीय राजनीति और जनभागीदारी / Indian Politics and Participation, Public Administration and Governance / लोक प्रशासन और सुशासन, Sociological Thinkers / समाजशास्त्रीय विचारक, Indian Society and Population / भारतीय समाज और जनसंख्या, Ethics / नीतिशास्त्र, Statement-Based Reasoning / कथन आधारित तार्किक विवेचन, State-Specific General Knowledge / राज्य विशेष सामान्य ज्ञान
  21. GS Paper | 07 | Current Events / समसामयिक घटनाएँ: Current Affairs / समसामयिकी, Teaching Methods and Approaches / शिक्षण विधियाँ एवं उपागम, Teacher: Qualities, Professional Ethics and Development / शिक्षक: गुण, व्यावसायिक नैतिकता एवं विकास, Assessment and Evaluation / आकलन एवं मूल्यांकन, International Economics / अंतर्राष्ट्रीय अर्थशास्त्र, International Relations: Theories and Concepts / अंतरराष्ट्रीय संबंध: सिद्धांत और अवधारणाएँ, India's Foreign Policy / भारत की विदेश नीति, Social Problems and Welfare / सामाजिक समस्याएँ और कल्याण, Agricultural Economics and Extension / कृषि अर्थशास्त्र एवं प्रसार शिक्षा, Olympic Movement, Sports Bodies and Awards / ओलंपिक आंदोलन, खेल संस्थाएँ एवं पुरस्कार, खेल एवं क्रीड़ाएँ / Games and Sports, Sports Management and Tournaments / खेल प्रबंधन एवं प्रतियोगिताएँ, Physical Education Pedagogy and Inclusion / शारीरिक शिक्षा का शिक्षणशास्त्र एवं समावेशन, Modern Indian Art / आधुनिक भारतीय कला, History of Indian Dance and Treatises / नृत्य का इतिहास और शास्त्रीय ग्रंथ, Cultural and Language Awareness / संस्कृति और भाषा परिचय, Defence and Security Awareness / रक्षा और सुरक्षा परिचय, राष्ट्रीय पहचान और महत्वपूर्ण दिवस / National Identity and Important Days, Books and Publications / पुस्तकें और प्रकाशन, Teaching-Learning Process and Classroom Management / शिक्षण-अधिगम प्रक्रिया एवं कक्षा प्रबंधन
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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This video is a comprehensive mathematics lecture on the concepts of multiples and factors, designed for competitive exam preparation. The instructor begins by defining factors and multiples using the number 12 as an example, showing that its factors are {1, 2, 3, 4, 6, 12} and its multiples are {12, 24, 36, 48, ...}. He then introduces the fundamental formula for finding the number of factors of a number N, which is derived from its prime factorization: if N = p^a * q^b * r^c, then the number of factors is (a+1)(b+1)(c+1). The lecture progresses to solve a series of practice problems, including finding the number of factors of a sum, determining the smallest number to multiply to make a perfect square or cube, and solving remainder problems using the formula Dividend = Divisor * Quotient + Remainder. The video concludes with a problem involving the sum of digits of a large number and a unique problem about the square root of a repeating digit sequence.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card for 'NUMBER SYSTEM' followed by a presentation slide titled 'MULTIPLES AND FACTORS'. The instructor, Yash Jain Sir, begins the lesson by defining factors of a number N as all numbers that divide N completely. He introduces the concept of the number of factors and the product of factors, setting the stage for the main topic.

  2. 2:00 – 5:00 02:00-05:00

    The instructor writes the title 'Concept of Multiples & Factors' on the whiteboard. He uses the number 12 as an example, writing '12 = {1, 2, 3, 4, 6, 12}' to list its factors. He then explains that the number of factors of 12 is 6 and that the product of its factors is 12^3, which is 1728. He also writes the formula for the number of factors: (a+1)(b+1)(c+1) for a number N = p^a * q^b * r^c.

  3. 5:00 – 10:00 05:00-10:00

    The instructor demonstrates the formula for the number of factors using the number 12. He shows the prime factorization of 12 as 2^2 * 3^1. Applying the formula, he calculates the number of factors as (2+1)(1+1) = 3 * 2 = 6. He then explains that the product of the factors of a number N is N^(number of factors / 2), which for 12 is 12^3. He also discusses the concept of infinite multiples and limited factors.

  4. 10:00 – 15:00 10:00-15:00

    The instructor presents a slide titled 'Number of Factors/Divisors' and reviews the basic formula for the number of factors of a number N, which is (a+1)(b+1)(c+1) where N = p^a * q^b * r^c. He then moves to a practice problem: 'The number A4531B where A & B are single digits, the given number is divisible by 72. Then the number of factors for A + B is'. He begins to solve it by factoring 72 as 2^3 * 3^2.

  5. 15:00 – 20:00 15:00-20:00

    The instructor continues solving the problem about the number A4531B. He determines that for the number to be divisible by 72, it must be divisible by both 8 and 9. He uses the divisibility rule for 8 (last three digits must be divisible by 8) to find that B must be 2. He then uses the divisibility rule for 9 (sum of digits must be divisible by 9) to find that A must be 3. He concludes that A + B = 5 and the number of factors of 5 is 2.

  6. 20:00 – 25:00 20:00-25:00

    The instructor presents a new problem: 'Let N = 162, let P be the smallest number multiplied to N such that N becomes a perfect square. Let Q be the smallest number multiplied to N such that N becomes a perfect cube. Find the product of factors of P+Q'. He begins by factoring 162 as 2 * 3^4. To make it a perfect square, he needs to multiply by 2, so P = 2. To make it a perfect cube, he needs to multiply by 2^2 * 3^2, so Q = 36.

  7. 25:00 – 30:00 25:00-30:00

    The instructor continues the problem with N=162. He calculates P = 2 and Q = 36, so P + Q = 38. He then needs to find the product of the factors of 38. He factors 38 as 2 * 19, so the number of factors is (1+1)(1+1) = 4. The product of the factors is 38^(4/2) = 38^2 = 1444. He then moves to the next problem.

  8. 30:00 – 35:00 30:00-35:00

    The instructor presents a problem: 'x is the smallest number such that x/2 is a perfect square and x/3 is a perfect cube. Find the number of divisors of x'. He sets up the equations x/2 = a^2 and x/3 = b^3. He then combines them to get x = 2a^2 = 3b^3. He deduces that x must be divisible by 2 and 3, so he lets x = 2^a * 3^b. He then uses the conditions to find that a must be odd and b must be even, and that a-1 must be divisible by 3 and b-1 must be divisible by 2.

  9. 35:00 – 40:00 35:00-40:00

    The instructor continues solving for x. He finds the smallest values for a and b that satisfy the conditions: a=3 and b=4. Therefore, x = 2^3 * 3^4 = 8 * 81 = 648. He then calculates the number of divisors of x as (3+1)(4+1) = 4 * 5 = 20.

  10. 40:00 – 45:00 40:00-45:00

    The instructor presents a new problem: 'A number is divided by 406 leaves remainder 115. What will be the remainder when it will be divided by 29?'. He writes the formula Dividend = Divisor * Quotient + Remainder, so n = 406q + 115. He then divides this expression by 29 to find the remainder. He calculates 406 / 29 = 14, so 406q is divisible by 29. He then finds the remainder of 115 / 29, which is 28.

  11. 45:00 – 50:00 45:00-50:00

    The instructor presents a problem: 'A number when divided by D leaves a remainder of 8 and when divided by 3D leaves a remainder of 21. What is the remainder left, when twice the number is divided by 3D?'. He sets up the equations n = Dx + 8 and n = 3Dy + 21. He equates them to get Dx + 8 = 3Dy + 21, which simplifies to D(x - 3y) = 13. He then considers the possible values for D and x-3y, finding D=13 and x-3y=1. He then finds n = 13*1 + 8 = 21, so 2n = 42. Finally, he divides 42 by 3D = 39 to get a remainder of 3.

  12. 50:00 – 55:00 50:00-55:00

    The instructor presents a problem: 'If n is a natural number such that 10^100 < n < 10^101 and the sum of the digits on n is 2, then the number of values n can take is'. He explains that n is a 101-digit number. The sum of its digits is 2, so the only possibilities are a 1 followed by 100 zeros, or a 2 followed by 100 zeros, or a 1 followed by a 1 and then 99 zeros. He calculates the number of such numbers as 100 (for 1 followed by 1 and 99 zeros) + 1 (for 2 followed by 100 zeros) + 1 (for 1 followed by 100 zeros) = 102.

  13. 55:00 – 60:00 55:00-60:00

    The instructor presents a problem: 'The square root of 12345678987654321 is nnnnnn... upto p times, find the sum of n and p?'. He shows that the square root of 121 is 11, the square root of 12321 is 111, and the square root of 1234321 is 1111. He identifies the pattern: the square root of a number that reads 123...n...321 is a number with n ones. The given number is 12345678987654321, which has 9 digits in the middle, so the square root is 111111111, which is 9 ones. Therefore, n=1 and p=9, so n+p=10.

  14. 60:00 – 65:00 60:00-65:00

    The instructor continues the problem about the square root of 12345678987654321. He confirms that the number is a palindrome and that its square root is a number with 9 ones, which is 111111111. He then states that the sum of n and p is 1 + 9 = 10. He then moves to the next problem.

  15. 65:00 – 70:00 65:00-70:00

    The instructor presents a problem: 'If n is a natural number such that 10^100 < n < 10^101 and the sum of the digits on n is 2, then the number of values n can take is'. He explains that n is a 101-digit number. The sum of its digits is 2, so the only possibilities are a 1 followed by 100 zeros, or a 2 followed by 100 zeros, or a 1 followed by a 1 and then 99 zeros. He calculates the number of such numbers as 100 (for 1 followed by 1 and 99 zeros) + 1 (for 2 followed by 100 zeros) + 1 (for 1 followed by 100 zeros) = 102.

  16. 70:00 – 75:00 70:00-75:00

    The instructor continues the problem about the number of values n can take. He explains that n is a 101-digit number. The sum of its digits is 2, so the only possibilities are a 1 followed by 100 zeros, or a 2 followed by 100 zeros, or a 1 followed by a 1 and then 99 zeros. He calculates the number of such numbers as 100 (for 1 followed by 1 and 99 zeros) + 1 (for 2 followed by 100 zeros) + 1 (for 1 followed by 100 zeros) = 102.

  17. 75:00 – 80:00 75:00-80:00

    The instructor presents a problem: 'The square root of 12345678987654321 is nnnnnn... upto p times, find the sum of n and p?'. He shows that the square root of 121 is 11, the square root of 12321 is 111, and the square root of 1234321 is 1111. He identifies the pattern: the square root of a number that reads 123...n...321 is a number with n ones. The given number is 12345678987654321, which has 9 digits in the middle, so the square root is 111111111, which is 9 ones. Therefore, n=1 and p=9, so n+p=10.

  18. 80:00 – 81:02 80:00-81:02

    The video ends with a closing screen showing the 'KG' logo for Knowledge Gate and the website www.knowledgegate.in. The background is a dark, abstract design with glowing orange lines, signifying the end of the lecture.

This video is a structured and comprehensive lecture on the topic of number systems, specifically focusing on multiples and factors. The instructor begins with foundational definitions and formulas, using the number 12 as a primary example to illustrate the concepts of factors, multiples, and the formula for the number of factors. The core of the video is a series of practice problems that progressively increase in complexity, covering topics such as divisibility rules, finding the smallest multiplier to form a perfect square or cube, and solving remainder problems using the fundamental division algorithm. The lecture also includes a unique problem involving the square root of a palindromic number, demonstrating a pattern recognition skill. The overall teaching style is methodical, with clear step-by-step solutions, making it an effective resource for students preparing for competitive exams.

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