Tricks to Find Factors that are Perfect Squares & Cubes

Duration: 11 min

This video lesson is available to enrolled students.

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Module outline

  1. About the Course: About the Course
  2. Section 1 | Language (Hindi): भाषा, लिपि और वर्ण, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, पद-भेद और रूप-विकार, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, संधि, समास, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, शब्द-भंडार, अपठित बोध, Multi-Topic Content (Cross-Concept Material), वर्णमाला
  3. Section 1 | Language (उर्दू भाषा — 22 प्रश्न (निश्चित)): رسم الخط، حروفِ تہجی اور املا / Script, Alphabet and Spelling, اسم، ضمیر اور صفت / Nouns, Pronouns and Adjectives, الفاظ، محاورے اور ضرب الامثال / Vocabulary, Idioms and Proverbs, تفہیمِ عبارت، انشا اور ترجمہ / Comprehension, Writing and Translation, Comprehension
  4. Section 1 | Language (বাংলা / बांग्ला भाषा — 22 प्रश्न (निश्चित)): শব্দার্থ, বাগধারা ও ভাষাশুদ্ধি / Meaning, Idioms and Correct Usage, বাক্য ও বাচ্য / Sentences and Voice, শব্দগঠন ও শব্দভান্ডার / Word Formation and Vocabulary, অনুবাদ, বোধপরীক্ষণ ও রচনা / Translation, Comprehension and Composition
  5. Section 1 | Language (English): Vocabulary, Noun, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Verbs and Tenses, Pronoun, Verb, Adverb, Prepositions and Conjunctions, Parts of Speech, Determiner, Conjunction, Modals, Transformation of Sentences, Direct and Indirect Speech, Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Sentence Structure and Common Errors, Reading Comprehension, Cloze Test (Cloze Passage), Verbal Analogy (Word Based Analogy), Facts, Inference and Judgements
  6. Section 2 | Primary Maths & Reasoning: अनुपात और समानुपात / Ratio and Proportion, Divisibility Rules, मिश्रण और एलिगेशन / Mixture and Alligation, Permutation and Combination, संख्या पद्धति / Number System, महत्तम समापवर्तक और लघुत्तम समापवर्त्य / HCF and LCM, Sequence and Series (Series), समय, चाल और दूरी / Time, Speed and Distance, श्रेणी / Series, कूट लेखन एवं कूट वाचन / Coding-Decoding, अशाब्दिक तर्क / Non-Verbal Reasoning, प्रतिशत / Percentage, औसत / Average, लाभ, हानि और बट्टा / Profit, Loss and Discount, साधारण और चक्रवृद्धि ब्याज / Simple and Compound Interest, समय और कार्य / Time and Work, घन क्षेत्रमिति / Solid Mensuration, Age Problems, Arithmetic, सरलीकरण, घातांक और करणी / Simplification, Indices and Surds, Algebra, Probability, Trigonometry, Syllogisms, बैठक व्यवस्था एवं पहेलियाँ / Seating Arrangement and Puzzles, रक्त संबंध / Blood Relations, दिशा एवं दूरी / Direction and Distance, Analytical Reasoning (Counting Figures Reasoning), गणितीय संक्रियाएँ एवं लुप्त संख्या / Mathematical Operations and Missing Numbers, कैलेंडर एवं घड़ी / Calendar and Clock, Clock, घन, पासा एवं आकृतियों की गिनती / Cubes, Dice and Figure Counting, वेन आरेख, न्याय वाक्य एवं असमानता / Venn Diagrams, Syllogism and Inequalities, Data Arrangement, Inequality, सादृश्यता एवं वर्गीकरण / Analogy and Classification, क्रम एवं रैंकिंग / Order and Ranking, समुच्चय, संबंध, फलन और विविक्त गणित / Sets, Relations, Functions and Discrete Mathematics, संख्या पद्धति और संख्या सिद्धांत / Number Systems and Number Theory, निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति / Coordinate Geometry, Vectors and 3D Geometry, बहुपद, समीकरण और सम्मिश्र संख्याएँ / Polynomials, Equations and Complex Numbers
  7. Section 2 | General Awareness (GK + Current Affairs): Polity, वैज्ञानिक उपलब्धियाँ और अनुसंधान / Scientific Achievements and Research, Biology, राज्य विशेष सामान्य ज्ञान / State-Specific General Knowledge, Chemistry, History, Geography, Indian Economy, Environment, समसामयिकी / Current Affairs, Static General Knowledge
  8. Section 3 | Computer System and Org. (CSO): Basics of COA, Floating Point Rep, Cache Memory Organization, Instr Formats & Modes, Programming the Basic Computer, Input Output Organisation, Register Transfer and Microoperations, Control Unit Design
  9. Section 3 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, SQL, File Organization & Indexing, Relational Algebra, Relational Calculus, Transaction Management, Concurrency Control
  10. Section 3 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Process Synchronization
  11. Section 3 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics
  12. Section 3 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Shortest Path Algos
  13. Section 3 | Theory Of Computation & Compiler Design: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization
  14. Section 3 | Emerging Trends (AI, Societal Ethics & MS Office): कंप्यूटर के मूल सिद्धांत / Computer Fundamentals, Emerging Tech Trends, Society, Law & Ethics, MS Office Applications, व्यापार, विपणन एवं उपभोक्ता संरक्षण / Trade, Marketing and Consumer Protection, साइबर सुरक्षा / Cyber Safety and Security, कंप्यूटर सॉफ़्टवेयर / Computer Software, ऑपरेटिंग सिस्टम / Operating System, कंप्यूटर नेटवर्क / Computer Networks, वर्ड प्रोसेसिंग (MS Word) / Word Processing (MS Word), Terms & Abbreviations, प्रेज़ेंटेशन (MS PowerPoint) / Presentation Software (MS PowerPoint)
  15. Section 3 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Maintenance and Configuration Management, Software Testing
  16. Section 3 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  17. Section 3 | OOPS, C++, JAVA, HTML: OOPS with C++, Java, HTML Basics, CSS
  18. Section 3 | Java: Java Fundamentals, Array, String, IO & References, OOPS with Java
  19. Section 3 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  20. Section 3 | Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  21. Section 3 | Multimedia: Multimedia
  22. BPSC Previous Year Papers:
  23. BPSC Full Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video is a mathematics lecture focused on number theory, specifically on finding the number and sum of factors of a given number that are perfect squares. The video begins with an introduction to the topic, followed by a detailed, step-by-step solution to a problem: 'Find the number and Sum of factors of 544 which are perfect squares?'. The instructor first performs the prime factorization of 544, which is 2^5 x 17^1. He then explains the method for finding factors that are perfect squares by considering only the even powers of the prime factors. For the prime 2, the even powers are 2^0, 2^2, and 2^4, and for 17, the only even power is 17^0. The number of such factors is calculated as (3) x (1) = 3. The sum of these factors is calculated as (1 + 4 + 16) x (1) = 21. The instructor verifies this result using an online Divisors Calculator, which confirms the total sum of all divisors is 114 and the number of divisors is 12. The video concludes by posing a follow-up question to find the sum of factors that are NOT perfect squares, which is solved by subtracting the sum of perfect square factors (21) from the total sum (114), resulting in 113. The video uses a digital whiteboard for all calculations and includes a small video feed of the instructor.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card displaying 'NUMBER SYSTEM' in a stylized font. This transitions to a presentation slide with the title 'NUMBER SYSTEM' and the subtitle 'The mysterious world of numbers...'. The instructor, Yash Jain, is visible in a small window in the bottom right corner. The slide also features the text 'Basic To Advance' and the logo 'YASH [KG]'. The instructor introduces the topic of the video, which is the number system, and the specific problem to be solved: 'Find the number and Sum of factors of 544 which are perfect squares?'. The background is a light orange with scattered geometric shapes and confetti-like patterns.

  2. 2:00 – 5:00 02:00-05:00

    The instructor begins solving the problem on a digital whiteboard. He starts by writing the number 544 and proceeds to find its prime factorization. He writes '544 = 32 x 17' and then '544 = 2^5 x 17'. He explains that to find factors that are perfect squares, we need to consider only the even powers of the prime factors. He then writes the formula for the sum of factors of a number, (p^(a+1)-1)/(p-1), and applies it to the prime 2, writing (2^(5+1)-1)/(2-1) and simplifying it to (64-1)/1 = 63. He then applies the same formula to the prime 17, writing (17^(1+1)-1)/(17-1) and simplifying it to (289-1)/16 = 288/16 = 18. He then multiplies these results to get the total sum of all factors, 63 x 18 = 1134. He then begins to explain how to find the sum of factors that are perfect squares.

  3. 5:00 – 10:00 05:00-10:00

    The instructor explains the method for finding the sum of factors that are perfect squares. He states that for a factor to be a perfect square, the power of each prime in its factorization must be even. For the prime 2, the even powers are 2^0, 2^2, and 2^4. For the prime 17, the only even power is 17^0. He writes the sum of these factors as (2^0 + 2^2 + 2^4) x (17^0). He calculates this as (1 + 4 + 16) x (1) = 21. He then writes the number of such factors as (3) x (1) = 3. He then uses an online Divisors Calculator to verify his results. The calculator shows that the divisors of 544 are 1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, 544, the number of divisors is 12, and the sum of divisors is 114. He notes that the sum of all divisors is 114, not 1134 as he previously calculated, indicating a mistake in his earlier calculation. He then corrects his calculation, stating that the sum of all divisors is 114 and the number of divisors is 12.

  4. 10:00 – 10:45 10:00-10:45

    The instructor poses a follow-up question: 'Find the number and Sum of factors of 544 which are NOT perfect squares?'. He explains that this can be found by subtracting the sum of perfect square factors from the total sum of all factors. He writes the equation: Sum = Total - Perfect Square = 114 - 21 = 113. He then writes the number of such factors as: Number = Total - Perfect Square = 12 - 3 = 9. The video ends with a 'Thank You for Watching' screen with a neon sign effect.

The video presents a clear, step-by-step tutorial on a specific number theory problem. It begins with a problem statement, proceeds through a detailed mathematical derivation using prime factorization and the formula for the sum of divisors, and concludes with a verification step using an online tool. The key learning point is the method for isolating factors that are perfect squares by restricting the exponents to even numbers. The instructor's process of identifying and correcting a calculation error (1134 vs 114) is a valuable lesson in mathematical rigor. The video effectively uses a digital whiteboard to illustrate the logic and calculations, making the abstract concepts accessible to students.

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