4 Jan - Aptitude - Number System

Duration: 1 hr 3 min

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AI Summary

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This video is a comprehensive lecture on number theory concepts for the GATE exam, presented by an instructor named Yash Jain. The lecture begins with an introduction to the topic of number systems and then systematically covers key concepts. The first major concept is 'Factors,' where the instructor explains how to find the number of factors of a number using its prime factorization, demonstrated with examples like 12 and 72. The second concept is the 'Magic of 9,' which is used to solve problems involving perfect squares and cubes, such as finding the smallest number to multiply 162 to make it a perfect cube. The lecture also includes a section on modular arithmetic, specifically calculating 3^51 mod 5, which is solved by identifying the repeating pattern of remainders. The video concludes with a discussion on the square root of a large number and a brief overview of the course structure on the KnowledgeGate platform, which includes various modules and topics for GATE preparation.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video starts with a black screen displaying the name 'Rajveer' in white text, which then changes to 'ALINA TURAK'. This appears to be an introductory title sequence for the video or a segment within it.

  2. 2:00 – 5:00 02:00-05:00

    The video transitions to a lecture on 'NUMBER SYSTEM'. The first concept, 'Concept 1 - Factors', is introduced. The instructor explains that factors of a number N are numbers that divide N completely. He demonstrates this with the number 12, listing its factors as {1, 2, 3, 4, 6, 12} and showing the multiplication pairs (1x12, 2x6, 3x4) that equal 12.

  3. 5:00 – 10:00 05:00-10:00

    The lecture continues with the concept of 'Number of Factors/Divisors'. A formula is presented: if N = p^a * q^b * r^c, then the number of factors of N is (a+1)(b+1)(c+1). The instructor applies this to the number 72, which has a prime factorization of 2^3 * 3^2, resulting in (3+1)(2+1) = 12 factors. He then applies the same formula to 12000, which is 2^5 * 3^1 * 5^3, to find it has (5+1)(1+1)(3+1) = 48 factors.

  4. 10:00 – 15:00 10:00-15:00

    The instructor presents a GATE CSE 2015 question: 'The number of divisors of 2100 is ____'. He first performs the prime factorization of 2100 as 2^2 * 3^1 * 5^2 * 7^1. Using the formula, he calculates the number of divisors as (2+1)(1+1)(2+1)(1+1) = 3 * 2 * 3 * 2 = 36. The answer is confirmed to be 36.

  5. 15:00 – 20:00 15:00-20:00

    A new question is introduced: 'The number of distinct positive integral factors of 2014 is ____'. The instructor begins the prime factorization of 2014, writing 2014 = 2 x 1007. He then factors 1007 as 19 x 53, concluding that 2014 = 2^1 * 19^1 * 53^1. He then applies the divisor formula: (1+1)(1+1)(1+1) = 2 * 2 * 2 = 8.

  6. 20:00 – 25:00 20:00-25:00

    The instructor presents a GATE CSE 2025 question about two arbitrary prime numbers p1 and p2. He evaluates the options by testing with specific values. For p1=2 and p2=3, he shows that p1+p2=5 (prime), p1*p2=6 (not prime), p1+p2+1=6 (not prime), and p1*p2+1=7 (prime). He eliminates options a, c, and d, concluding that b is correct: p1*p2 is not a prime number.

  7. 25:00 – 30:00 25:00-30:00

    The video shows a GATE CS 2017 question: 'Find the smallest number y such that y x 162 is a perfect cube'. The instructor performs the prime factorization of 162 as 2^1 * 3^4. To make it a perfect cube, all exponents must be multiples of 3. He determines that y must provide 2^2 and 3^2, so y = 2^2 * 3^2 = 4 * 9 = 36. The answer is D. 36.

  8. 30:00 – 35:00 30:00-35:00

    The instructor discusses a problem involving perfect squares. He writes the equation y * 162 = perfect square. He factors 162 as 2^1 * 3^4. For the product to be a perfect square, all exponents in the prime factorization must be even. Since the exponent of 2 is 1 (odd), y must provide at least one more factor of 2. The smallest such y is 2, making the product 2^2 * 3^4, which is a perfect square.

  9. 35:00 – 40:00 35:00-40:00

    A GATE CSE 2019 question is presented: 'The value of 3^51 mod 5 is ____'. The instructor begins by calculating the powers of 3 modulo 5: 3^1 mod 5 = 3, 3^2 mod 5 = 9 mod 5 = 4, 3^3 mod 5 = 27 mod 5 = 2, 3^4 mod 5 = 81 mod 5 = 1. He identifies a repeating cycle of length 4: 3, 4, 2, 1.

  10. 40:00 – 45:00 40:00-45:00

    The instructor continues solving the 3^51 mod 5 problem. He uses the cycle length of 4 to find the remainder. He divides 51 by 4, getting a quotient of 12 and a remainder of 3. This means 3^51 mod 5 is the same as 3^3 mod 5, which is 27 mod 5 = 2. The answer is 2.

  11. 45:00 – 50:00 45:00-50:00

    The video shows a problem: 'The square root of 12345678987654321 is nnnnn... up to p times, find the sum of n and p?'. The instructor writes the square root symbol over the number 12345678987654321 and notes that the result is a number with a repeating digit 'n' repeated 'p' times. He then lists three concepts: 1. factors, 2. remainder, 3. unit digit, suggesting these are the tools to solve the problem.

  12. 50:00 – 55:00 50:00-55:00

    The instructor shows a solution to a problem involving the number 2014. He states that 2014 = 2 x 1007 = 2 x 19 x 53. He then applies the divisor formula: (1+1)(1+1)(1+1) = 8. He also mentions the 'Remainder Theorem' and writes a formula for the sum of digits: a+b+c+d = a/4 + b/4 + c/4 + d/4, which is a misapplication of the theorem.

  13. 55:00 – 60:00 55:00-60:00

    The instructor continues to solve the 3^51 mod 5 problem. He has established the cycle: 3^1 mod 5 = 3, 3^2 mod 5 = 4, 3^3 mod 5 = 2, 3^4 mod 5 = 1. He then calculates 51 divided by 4, which gives a remainder of 3. Therefore, 3^51 mod 5 is the same as 3^3 mod 5, which is 27 mod 5 = 2. The answer is 2.

  14. 60:00 – 63:06 60:00-63:06

    The video shows a screen recording of the KnowledgeGate website. The instructor navigates through the 'GATE Guidance by Sanchit Sir' course, which has 16 modules and 155 topics. He scrolls through the course contents, showing topics like 'Speed and Distance', 'Profit and Loss', 'Powers and Exponents', and 'Deductive and Inductive Reasoning'. The interface shows a 'Goal Slider' and 'Your Progress' for each topic.

This video is a structured lecture on number theory, designed for GATE exam preparation. It systematically progresses from fundamental concepts to complex problem-solving. The core of the lesson revolves around two key ideas: the formula for the number of factors of a number based on its prime factorization, and the use of modular arithmetic to find remainders. The instructor demonstrates these concepts by solving a series of GATE exam questions, including finding the number of divisors, identifying properties of prime numbers, and calculating perfect powers. The video concludes with a practical demonstration of the course platform, reinforcing its purpose as an educational resource.