Concept of Reminders (Part 4)

Duration: 18 min

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Inside: a video lesson and guided study material.

Module outline

  1. Paper - 1 | Unit - 1 | Teaching Aptitude: शिक्षण की अवधारणा एवं उद्देश्य / Concept and Objectives of Teaching, शिक्षण के स्तर / Levels of Teaching, शिक्षार्थी की विशेषताएँ / Learner's Characteristics, अधिगम: प्रकृति एवं सिद्धांत / Learning: Nature and Theories, संज्ञान, स्मृति एवं अधिसंज्ञान / Cognition, Memory and Metacognition, शिक्षण को प्रभावित करने वाले कारक / Factors Affecting Teaching, प्रभावी शिक्षण एवं शिक्षक / Effective Teaching and the Teacher, अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन / Instructional Objectives and Lesson Planning, शिक्षण विधियाँ / Methods of Teaching, ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ / Offline and Online Teaching Methods, शिक्षण सहायक प्रणाली / Teaching Support System, मूल्यांकन प्रणाली / Evaluation Systems, Sāṃkhya, व्यक्तिगत व्यवहार / Individual Behaviour, मानव संसाधन प्रबंधन / Human Resource Management, बाल्यावस्था में विकास के क्षेत्र / Domains of Child Development, संवेग का जैविक आधार / Biological Basis of Emotion, उपनिषद और भगवद्गीता / Upanishads and Bhagavad Gita, केन्द्रीय और परिधीय तंत्रिका तंत्र / Central and Peripheral Nervous Systems
  2. Paper - 1 | Unit - 2 | Research Aptitude: अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि / Research: Meaning, Features and Scientific Method, अनुसंधान के प्रतिमान एवं उपागम / Research Paradigms and Approaches, अनुसंधान के प्रकार / Types of Research, चर एवं मापन / Variables and Measurement, अनुसंधान की विधियाँ / Methods of Research, अनुसंधान के चरण / Steps of Research, प्रतिचयन / Sampling, आँकड़ा संग्रह के उपकरण एवं विधियाँ / Tools and Methods of Data Collection, आँकड़ा विश्लेषण एवं शोध सांख्यिकी / Data Analysis and Statistics in Research, शोध-प्रबंध एवं लेख लेखन / Thesis and Article Writing, अनुसंधान में ICT का अनुप्रयोग / Application of ICT in Research, शोध नैतिकता / Research Ethics
  3. Paper - 1 | Unit - 3 | Comprehension: प्रत्यक्ष बोध / Literal Comprehension, गद्यांश में शब्दावली और भाषा / Vocabulary and Language in Context, निष्कर्षात्मक बोध / Inferential Comprehension, आलोचनात्मक बोध / Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Restoration and Eighteenth-Century Drama, Modern British Fiction, William Shakespeare, Mid-Twentieth-Century and Contemporary British Fiction, American Drama
  4. Paper - 1 | Unit - 4 | Communication: संचार: अर्थ, प्रकृति एवं प्रक्रिया / Communication: Meaning, Nature and Process, संचार के प्रकार एवं स्तर / Types and Levels of Communication, मौखिक एवं अमौखिक संचार / Verbal and Non-verbal Communication, संकेत विज्ञान एवं अर्थ / Semiotics and Meaning, संचार के मॉडल एवं सिद्धांत / Models and Theories of Communication, प्रभावी संचार / Effective Communication, संचार की बाधाएँ / Barriers to Communication, कक्षा संचार / Classroom Communication, जनसंचार माध्यम: प्रकार एवं विकास / Mass Media: Types and Evolution, जनसंचार एवं समाज / Mass Media and Society, डिजिटल एवं नव माध्यम / Digital and New Media, Static General Knowledge, Morphology and Syntax, जन स्वास्थ्य पोषण / Public Health Nutrition, Newtonian Motion and Stability, GK- Current Affairs, Indian Economy, Art and Architecture, India after Independence
  5. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: संख्या पद्धति एवं सरलीकरण / Number System and Simplification, बीजगणित एवं आयु / Algebra and Ages, प्रतिशतता / Percentage, अनुपात, समानुपात, साझेदारी एवं मिश्रण / Ratio, Proportion, Partnership and Mixtures, औसत / Averages, लाभ, हानि एवं बट्टा / Profit, Loss and Discount, साधारण एवं चक्रवृद्धि ब्याज / Simple and Compound Interest, समय एवं कार्य / Time and Work, चाल, समय एवं दूरी / Speed, Time and Distance, क्षेत्रमिति एवं ज्यामिति / Mensuration and Geometry, क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, श्रेणी / Series, कूटलेखन-कूटवाचन / Coding-Decoding, संबंध, दिशा एवं क्रम / Relationships, Directions and Ranking, व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर / Arrangements, Puzzles, Clock and Calendar, Mathematical Foundations, Polity, जीवनी, Deconstruction, Gender and Queer Theory, Modern British and Irish Poetry, Central Force and Scattering, Chemical thermodynamics, Microscopic techniques, संविधान का दर्शन / Philosophy of the Constitution, Waves, Optics and Charged-Particle Motion, Membrane structure and function, Analysis, Markov Chains, Univariate and Joint Distributions
  6. Paper - 1 | Unit - 6 | Logical Reasoning: युक्ति की संरचना / Structure of Arguments, पद, परिभाषा एवं भाषा के उपयोग / Terms, Definitions and Uses of Language, निरुपाधिक प्रतिज्ञप्तियाँ / Categorical Propositions, निरुपाधिक न्यायवाक्य / Categorical Syllogism, वेन आरेख / Venn Diagrams, निगमन, आगमन एवं सादृश्यता / Deduction, Induction and Analogy, तर्कदोष / Fallacies, भारतीय तर्कशास्त्र: प्रमाण / Indian Logic: Pramanas, अनुमान एवं व्याप्ति / Anumana and Vyapti, हेत्वाभास / Hetvabhasa, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Aristotle, पाश्चात्य दर्शन की विचारधाराएँ / Western Schools of Philosophy, Kant, Sadhana Paada, Vaiśeṣika
  7. Paper - 1 | Unit - 7 | Data Interpretation: आँकड़े एवं आलेखीय निरूपण / Data and Graphical Representation, आँकड़ा निर्वचन: तालिका / Data Interpretation: Tables
  8. Paper - 1 | Unit - 8 | Information and Communication Technology: कंप्यूटर की मूल बातें और हार्डवेयर / Computer Fundamentals and Hardware, मेमोरी, भंडारण और डेटा निरूपण / Memory, Storage and Data Representation, कंप्यूटर सॉफ्टवेयर / Computer Software, ऑफिस अनुप्रयोग / Office Applications, मल्टीमीडिया और फ़ाइल प्रारूप / Multimedia and File Formats, कंप्यूटर नेटवर्क / Computer Networks, इंटरनेट और वर्ल्ड वाइड वेब / Internet and World Wide Web, ई-मेल और ऑनलाइन संचार / Email and Online Communication, साइबर सुरक्षा / Cyber Security, डिजिटल नैतिकता, बौद्धिक संपदा और समाज / Digital Ethics, IPR and Society, क्लाउड और वितरित कंप्यूटिंग / Cloud and Distributed Computing, उभरती प्रौद्योगिकियाँ / Emerging Technologies, शिक्षा और शासन में ICT / 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, Basics of COA, Introduction to OS, Introduction to CN, Data Communication, Net Layer: IP Addressing, Network Security, Number System, Number Representation, Society, Law & Ethics, Keyboard Shortcuts, Website, Browser & Services, भौगोलिक संकेतों का संरक्षण / Protection of Geographical Indications
  9. Paper - 1 | Unit - 9 | People Development and Environment: पारितंत्र और जैव विविधता / Ecosystems and Biodiversity, मानव और पर्यावरण की अंतःक्रिया / Human and Environment Interaction, विकास और पर्यावरण / Development and Environment, प्राकृतिक संसाधन: वन, जल और मृदा / Natural Resources – Forests, Water and Soil, ऊर्जा संसाधन / Energy Resources, वायु प्रदूषण / Air Pollution, जल प्रदूषण और उपचार / Water Pollution and Treatment, ध्वनि प्रदूषण / Noise Pollution, अपशिष्ट प्रबंधन / Waste Management, पर्यावरणीय स्वास्थ्य और विषविज्ञान / Environmental Health and Toxicology, जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ / Climate Change and Global Environmental Issues, प्राकृतिक प्रकोप और आपदा प्रबंधन / Natural Hazards and Disaster Management, भारतीय पर्यावरण कानून और नीतियाँ / Indian Environmental Laws and Policies, अंतर्राष्ट्रीय पर्यावरण समझौते / International Environmental Agreements, Environment, Geography, कारखाना और उद्योग व्यवस्थाएँ / Factory and Industry Systems, Earth Materials, Surface Features and Processes, संक्रामक रोग और प्रतिरक्षण / Communicable Diseases and Immunization
  10. Paper - 1 | Unit - 10 | Higher Education System: प्राचीन भारत में शिक्षा / Education in Ancient India, प्राचीन भारत के उच्च शिक्षा केंद्र / Ancient Centres of Higher Learning, औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग / Colonial Education Policies and Commissions, औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक / Colonial Colleges, Universities and Reformers, स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ / Education Commissions and Policies after Independence, राष्ट्रीय शिक्षा नीति 2020 / National Education Policy 2020, उच्च शिक्षा संस्थानों के प्रकार / Types of Higher Education Institutions, अधिगम कार्यक्रम और अकादमिक क्रेडिट / Learning Programmes and Academic Credits, व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा / Professional, Technical and Skill-Based Education, मूल्य शिक्षा और पर्यावरण शिक्षा / Value Education and Environmental Education, शासन, प्रशासन और वित्तपोषण / Governance, Administration and Financing, उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग / Regulation, Accreditation and Ranking
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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video, presented by Yash Jain from Knowledge Gate, is a comprehensive tutorial on solving remainder problems in number systems, a common topic in competitive exams. The video begins with an introduction to the concept of remainders, using a simple division example (7 ÷ 2 = 3 R1) to establish the foundational idea. The main content consists of two complex problems. The first problem involves the sum of the reciprocals of the first 13 natural numbers, expressed as a fraction x/13!. The instructor demonstrates that to find the remainder when x is divided by 11, one must analyze the numerator of the sum after a common denominator is applied. The key insight is that for any term where the denominator is not divisible by 11, the corresponding numerator term will be divisible by 11, leaving only the term where the denominator is 11 as the only non-zero contributor to the remainder. The second problem asks for the remainder when N = (1! + 2! + 3! + ... + 1000!)^40 is divided by 10. The instructor explains that for n ≥ 5, n! is divisible by 10, so all terms from 5! onwards are 0 modulo 10. This simplifies the sum to (1! + 2! + 3! + 4!)^40 = (1 + 2 + 6 + 24)^40 = 33^40. The final step is to find the remainder of 33^40 when divided by 10, which is equivalent to finding the last digit of 33^40. The video concludes with a summary of the problem-solving techniques, emphasizing the use of modular arithmetic and the simplification of large expressions by identifying terms that become zero under the given modulus.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card for 'NUMBER SYSTEM' and then transitions to a presentation slide. The slide features the title 'NUMBER SYSTEM' and the tagline 'The mysterious world of numbers...'. The instructor, Yash Jain, is visible in a small window. He introduces the topic of the 'Concept of Remainders' and provides a simple example on the board: '7 ÷ 2 = 3 R1', with an arrow pointing to the '1' and labeling it 'Remainder'. This establishes the basic definition of a remainder as the leftover part of a division.

  2. 2:00 – 5:00 02:00-05:00

    The video presents the first problem: 'The expression sigma(1/n) from n=1 to 13 can also be written as x/13!. What would be the remainder if x is divided by 11?'. The instructor begins to solve this by writing the sum as 1/1 + 1/2 + 1/3 + ... + 1/13. He then multiplies the entire sum by 13! to get x, which results in the expression x = 13!/1 + 13!/2 + 13!/3 + ... + 13!/13. He explains that to find x mod 11, he needs to evaluate each term in the sum modulo 11.

  3. 5:00 – 10:00 05:00-10:00

    The instructor analyzes the terms of the sum x = 13!/1 + 13!/2 + ... + 13!/13 modulo 11. He explains that 13! is divisible by 11, so 13! ≡ 0 (mod 11). For any term where the denominator is not 11, the numerator 13! is divisible by 11, making the entire term 0 modulo 11. The only term that is not divisible by 11 is the one where the denominator is 11, which is 13!/11. He simplifies this term to 13 × 12 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 13 × 12 × 10!. He then states that the remainder is the value of 13 × 12 × 10! modulo 11, which he will calculate in the next step.

  4. 10:00 – 15:00 10:00-15:00

    The instructor calculates the remainder of 13 × 12 × 10! when divided by 11. He uses the property that (p-1)! ≡ -1 (mod p) for a prime p (Wilson's Theorem). Since 11 is prime, 10! ≡ -1 (mod 11). He then simplifies 13 × 12 modulo 11: 13 ≡ 2 (mod 11) and 12 ≡ 1 (mod 11). Therefore, the expression becomes 2 × 1 × (-1) = -2. Since the remainder must be positive, he adds 11 to get 9. The final answer for the first problem is 9. He then transitions to the second problem.

  5. 15:00 – 18:29 15:00-18:29

    The second problem is introduced: 'What is the remainder when N = (1! + 2! + 3! + ... + 1000!)^40 is divided by 10?'. The instructor explains that for n ≥ 5, n! is divisible by 10 (since it contains both 2 and 5 as factors), so n! ≡ 0 (mod 10). This means all terms from 5! to 1000! are 0 modulo 10. The sum simplifies to (1! + 2! + 3! + 4!)^40 = (1 + 2 + 6 + 24)^40 = 33^40. The problem now is to find 33^40 mod 10, which is the last digit of 33^40. He notes that the last digit of 33 is 3, so he needs to find the last digit of 3^40. He then calculates the pattern of the last digit of powers of 3: 3^1=3, 3^2=9, 3^3=7, 3^4=1, and the cycle repeats every 4. Since 40 is divisible by 4, the last digit is 1. The final answer is 1. The video ends with a 'Thank You for Watching' screen.

The video provides a clear and structured lesson on applying modular arithmetic to solve complex remainder problems. It begins with a fundamental concept and progresses to two distinct, challenging problems. The first problem demonstrates the power of simplification by identifying that only one term in a large sum contributes to the remainder when the modulus is a prime number, leveraging Wilson's Theorem. The second problem illustrates the technique of reducing a sum by recognizing that terms beyond a certain point become zero modulo the divisor. The core teaching method is the step-by-step breakdown of complex expressions into manageable parts, using properties of factorials and modular arithmetic to arrive at the solution. The progression from a simple example to two advanced problems effectively builds the viewer's problem-solving skills for competitive exams.

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