Solving Linear Diophantine Equation with the Euclidean Algorithm

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  • Опубліковано 14 чер 2024
  • Solving Linear Diophantine Equation with the Euclidean Algorithm
    Explore the fascinating world of solving linear Diophantine equations using the powerful Euclidean Algorithm! Join us on a journey through the step-by-step process of finding integer solutions to these intriguing mathematical puzzles. Unlock the secrets of Diophantine equations and elevate your problem-solving skills with this comprehensive tutorial. #lineardiophantineequation #euclideanalgorithm #mathematics #mathtutorial #problemsolving #algebra
    Topics Covered:
    1. Understanding the basics of Linear Diophantine equation in integers.
    2. Analyzing the unique method of Euclidean Algorithm.
    3. Step-by-step approach to solving equation for integers.
    4. Tips and tricks for solving linear Diophantine equations
    5. Particular and general solutions.
    Timestamps:
    0:00 Introduction
    0:45 Steps of linear Diophantine equation
    1:48 GCD
    3:02 Euclidean algorithm
    6:44 Particular solution
    8:20 General solutions
    🎯 This video is perfect for students, math enthusiasts, or anyone seeking to sharpen their problem-solving skills and gain confidence in dealing with radical equations. 🎓📈
    🔔 Challenge yourself and see if you can solve the linear Diophantine equation before we do! Hit the like button if you're up for the challenge and remember to subscribe for more exhilarating math content! 🛎️🔔
    Don't forget to like, comment, and subscribe to join our math-loving community. Let's get started on this exciting journey together! 🤝🌟
    Thanks for Watching!
    @infyGyan

КОМЕНТАРІ • 10

  • @mohammedsaysrashid3587
    @mohammedsaysrashid3587 Місяць тому

    Super wonderful explanation...thanks for sharing

    • @infyGyan
      @infyGyan  Місяць тому

      Thanks for watching 🙏

  • @alibhukoo5400
    @alibhukoo5400 Місяць тому +2

    (x,y) =(3,-7),(12,-29),(21,-51),(30,-73),(39,-95),(48,-117)= any combo like x number(first number of x is 3 then always add 9) and for value of( y is then first number is -7 then always add -22)

    • @alibhukoo5400
      @alibhukoo5400 Місяць тому +1

      x=(9n-6) and y=(15-22n) where n is the number of terms

  • @mircoceccarelli6689
    @mircoceccarelli6689 Місяць тому +1

    👍👍👍
    66x + 27y = 9
    ( x , y ) = ( - 6 + 9k , 15 - 22k )
    = ( - 6 , 15 ) + k ( 9 , - 22 )
    k € Z
    😊🤪👋

  • @allozovsky
    @allozovsky Місяць тому +1

    But why haven't you divided the whole equation by gcd(66, 27) = 3 right off at the very beginning of the solution? That would have given us a much simpler (but equivalent) form of the original equation: *22x + 9y = 3,* and we would have gotten exactly the same set of solutions anyway.

    • @allozovsky
      @allozovsky Місяць тому

      But the presented solution is absolutely correct up to every tiny detail, of course. 👍

    • @infyGyan
      @infyGyan  Місяць тому +1

      Yes we can take that way too with gcd 1.
      Thanks 👍

  • @user-ji5su2uq9m
    @user-ji5su2uq9m Місяць тому

    another approach
    66x + 27y = 9 => 22x + 9y = 3 => x ≡ 0 (mod 3)
    let x = 3k, 66k + 9y = 3 => 22k + 3y = 1 => k ≡ 1 (mod 3)
    let k = 3n +1, 22(3n +1) + 3y = 1, 22n + y + 7 = 0 => y = -7 - 22n
    22*3k*(3n + 1) + 3*(-7 - 22n ) = 1
    66kn + 22k - 21 - 66n = 1 => 3kn + k - 3n - 1 = (k - 1)*(3n + 1) = 0
    so k = 1, x = 3k*(3n + 1) = 3 + 9n, y = -7 - 22n
    (x,y) =(3 + 9n, -7 - 22n) which is same as (-6 +9k, 15 - 22k) with n = k -1

    • @infyGyan
      @infyGyan  Місяць тому

      Thanks for sharing!!