Number System (Quick Revision & Practice Questions)

Duration: 1 hr 21 min

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Inside: a video lesson and guided study material.

Module outline

  1. Paper - 1 | Unit - 1 | Teaching Aptitude: Concept and Objectives of Teaching / शिक्षण की अवधारणा एवं उद्देश्य, Levels of Teaching / शिक्षण के स्तर, Learner's Characteristics / शिक्षार्थी की विशेषताएँ, Learning: Nature and Theories / अधिगम: प्रकृति एवं सिद्धांत, Cognition, Memory and Metacognition / संज्ञान, स्मृति एवं अधिसंज्ञान, Factors Affecting Teaching / शिक्षण को प्रभावित करने वाले कारक, Effective Teaching and the Teacher / प्रभावी शिक्षण एवं शिक्षक, Instructional Objectives and Lesson Planning / अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन, Methods of Teaching / शिक्षण विधियाँ, Offline and Online Teaching Methods / ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ, Teaching Support System / शिक्षण सहायक प्रणाली, Evaluation Systems / मूल्यांकन प्रणाली
  2. Paper - 1 | Unit - 2 | Research Aptitude: Research: Meaning, Features and Scientific Method / अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि, Research Paradigms and Approaches / अनुसंधान के प्रतिमान एवं उपागम, Types of Research / अनुसंधान के प्रकार, Variables and Measurement / चर एवं मापन, Methods of Research / अनुसंधान की विधियाँ, Steps of Research / अनुसंधान के चरण, Sampling / प्रतिचयन, Tools and Methods of Data Collection / आँकड़ा संग्रह के उपकरण एवं विधियाँ, Data Analysis and Statistics in Research / आँकड़ा विश्लेषण एवं शोध सांख्यिकी, Thesis and Article Writing / शोध-प्रबंध एवं लेख लेखन, Application of ICT in Research / अनुसंधान में ICT का अनुप्रयोग, Research Ethics / शोध नैतिकता
  3. Paper - 1 | Unit - 3 | Comprehension: Literal Comprehension, Vocabulary and Language in Context, Inferential Comprehension, Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions)
  4. Paper - 1 | Unit - 4 | Communication: Communication: Meaning, Nature and Process / संचार: अर्थ, प्रकृति एवं प्रक्रिया, Types and Levels of Communication / संचार के प्रकार एवं स्तर, Verbal and Non-verbal Communication / मौखिक एवं अमौखिक संचार, Semiotics and Meaning / संकेत विज्ञान एवं अर्थ, Models and Theories of Communication / संचार के मॉडल एवं सिद्धांत, Effective Communication / प्रभावी संचार, Barriers to Communication / संचार की बाधाएँ, Classroom Communication / कक्षा संचार, Mass Media: Types and Evolution / जनसंचार माध्यम: प्रकार एवं विकास, Mass Media and Society / जनसंचार एवं समाज, Digital and New Media / डिजिटल एवं नव माध्यम, General Awareness (Legacy Paper 1) / सामान्य जागरूकता (पुराना पेपर 1)
  5. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: Number System and Simplification / संख्या पद्धति एवं सरलीकरण, Algebra and Ages / बीजगणित एवं आयु, Percentage / प्रतिशतता, Ratio, Proportion, Partnership and Mixtures / अनुपात, समानुपात, साझेदारी एवं मिश्रण, Averages / औसत, Profit, Loss and Discount / लाभ, हानि एवं बट्टा, Simple and Compound Interest / साधारण एवं चक्रवृद्धि ब्याज, Time and Work / समय एवं कार्य, Speed, Time and Distance / चाल, समय एवं दूरी, Mensuration and Geometry / क्षेत्रमिति एवं ज्यामिति, Series / श्रेणी, Coding-Decoding / कूटलेखन-कूटवाचन, Relationships, Directions and Ranking / संबंध, दिशा एवं क्रम, Arrangements, Puzzles, Clock and Calendar / व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर, Mathematical Foundations, Analysis
  6. Paper - 1 | Unit - 6 | Logical Reasoning: Structure of Arguments / युक्ति की संरचना, Terms, Definitions and Uses of Language / पद, परिभाषा एवं भाषा के उपयोग, Categorical Propositions / निरुपाधिक प्रतिज्ञप्तियाँ, Categorical Syllogism / निरुपाधिक न्यायवाक्य, Venn Diagrams / वेन आरेख, Deduction, Induction and Analogy / निगमन, आगमन एवं सादृश्यता, Fallacies / तर्कदोष, Indian Logic: Pramanas / भारतीय तर्कशास्त्र: प्रमाण, Anumana and Vyapti / अनुमान एवं व्याप्ति, Hetvabhasa / हेत्वाभास, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Aristotle / अरस्तू, Western Schools of Philosophy / पाश्चात्य दर्शन की विचारधाराएँ, Kant / कांट, Sadhana Paada / साधनपाद, Vaiśeṣika / वैशेषिक
  7. Paper - 1 | Unit - 7 | Data Interpretation: Data and Graphical Representation / आँकड़े एवं आलेखीय निरूपण, Data Interpretation: Tables / आँकड़ा निर्वचन: तालिका
  8. Paper - 1 | Unit - 8 | Information and Communication Technology: Computer Fundamentals and Hardware, Memory, Storage and Data Representation, Computer Software, Office Applications, Multimedia and File Formats, Computer Networks, Internet and World Wide Web, Email and Online Communication, Cyber Security, Digital Ethics, IPR and Society, Cloud and Distributed Computing, Emerging Technologies, ICT in Education and Governance, Cyber Threats and Malware Attacks, Role of Internet and Web Services, Electronic Data Interchange and E-Commerce, Internet Fundamentals, Web Design & Development, Web Publishing & Hosting, Emerging Technologies, Terms & Abbreviations, Society, Law & Ethics, Keyboard Shortcuts, Website, Browser & Services, Protection of Geographical Indications / भौगोलिक संकेतों का संरक्षण
  9. Paper - 1 | Unit - 9 | People Development and Environment: Ecosystems and Biodiversity / पारितंत्र और जैव विविधता, Human and Environment Interaction / मानव और पर्यावरण की अंतःक्रिया, Development and Environment / विकास और पर्यावरण, Natural Resources – Forests, Water and Soil / प्राकृतिक संसाधन: वन, जल और मृदा, Energy Resources / ऊर्जा संसाधन, Air Pollution / वायु प्रदूषण, Water Pollution and Treatment / जल प्रदूषण और उपचार, Noise Pollution / ध्वनि प्रदूषण, Waste Management / अपशिष्ट प्रबंधन, Environmental Health and Toxicology / पर्यावरणीय स्वास्थ्य और विषविज्ञान, Climate Change and Global Environmental Issues / जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ, Natural Hazards and Disaster Management / प्राकृतिक प्रकोप और आपदा प्रबंधन, Indian Environmental Laws and Policies / भारतीय पर्यावरण कानून और नीतियाँ, International Environmental Agreements / अंतर्राष्ट्रीय पर्यावरण समझौते, Environment / पर्यावरण, Geography / भूगोल, Factory and Industry Systems / कारखाना और उद्योग व्यवस्थाएँ, Earth Materials, Surface Features and Processes, Communicable Diseases and Immunization / संक्रामक रोग और प्रतिरक्षण
  10. Paper - 1 | Unit - 10 | Higher Education System: Education in Ancient India / प्राचीन भारत में शिक्षा, Ancient Centres of Higher Learning / प्राचीन भारत के उच्च शिक्षा केंद्र, Colonial Education Policies and Commissions / औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग, Colonial Colleges, Universities and Reformers / औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक, Education Commissions and Policies after Independence / स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ, National Education Policy 2020 / राष्ट्रीय शिक्षा नीति 2020, Types of Higher Education Institutions / उच्च शिक्षा संस्थानों के प्रकार, Learning Programmes and Academic Credits / अधिगम कार्यक्रम और अकादमिक क्रेडिट, Professional, Technical and Skill-Based Education / व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा, Value Education and Environmental Education / मूल्य शिक्षा और पर्यावरण शिक्षा, Governance, Administration and Financing / शासन, प्रशासन और वित्तपोषण, Regulation, Accreditation and Ranking / उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग, Indian Polity and Governance (Legacy Unit 10) / भारतीय राजव्यवस्था और शासन (पुराना यूनिट 10)
  11. Paper 2 | Unit 1 | Discrete Structures and Optimization: Propositional and Predicate Logic, Set Theory, Relations, Functions, Permutation, Combination and Probability / क्रमचय, संचय एवं प्रायिकता, Graph Theory, Group Theory, Digital Systems & Boolean Basics, Boolean Expression, Boolean Minimization, Optimization, Markov Chains
  12. Paper 2 | Unit 2 | Computer System Architecture: Logic Gates & Hardware, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Basics of COA, Register Transfer and Microoperations, Programming the Basic Computer, Instr Formats & Modes, Control Unit Design, Pipelining, Input Output Organisation, Cache Memory Organization, Multiprocessors
  13. Paper 2 | Unit 3 | Programming Languages and Computer Graphics: Language Design, C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, HTML Basics, XML, JavaScript-Basics, Java, Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing, OOPS with C++, Java Fundamentals, Array, String, IO & References, OOPS with Java
  14. Paper 2 | Unit 4 | Database Management Systems: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, ORDBMS, Database Security & Authorization, Query Processing & Optimization, Enhanced Data Models, Data Warehousing & Mining, Big Data Systems, NoSQL
  15. Paper 2 | Unit 5 | System Software and 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, Linux OS, Security, Distributed Systems, Virtual Machines
  16. Paper 2 | Unit 6 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  17. Paper 2 | Unit 7 | Data Structures and Algorithms: 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, Advanced Algorithms
  18. Paper 2 | Unit 8 | Theory of Computation and Compilers: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  19. Paper 2 | Unit 9 | Data Communication and 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, Network Security, Cloud Computing, Mobile Computing
  20. Paper 2 | Unit 10 | Artificial Intelligence: Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Fuzzy Sets, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms
  21. Full Notes: Paper 1 Full Notes
  22. Paper 1 | Full Mock Tests:
  23. Paper 2 | Full Mock Tests:
  24. Paper 1 | Previous Year Papers:
  25. Paper 2 | Previous Year Papers:
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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