Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

ParthKohli

  • 2 years ago

Please explain me the Euclidean Algorithm. Help appreciated.

  • This Question is Closed
  1. KingGeorge
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 6

    First step, look at an example. I'll color things to make it easy to see what I'm doing.

  2. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Okay - I follow.

  3. KingGeorge
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 6

    Look at \(a=572\) and \(b=165\). The euclidean algorithm will give us the gcd of these two numbers. Here's how you would find it \[572=\color{red}{165}\cdot3+\color{green}{77}\]\[\color{red}{165}=\color{green}{77}\cdot2+\color{blue}{11}\]\[\color{green}{77}=\color{blue}{11}\cdot7+0\]Now that that last number is 0, look at the remainder above it. In this case, that's 11. Hence, \(\gcd(572,165)=11.\)

  4. KingGeorge
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 6

    The last number in each of those lines is called the "remainder" and the black numbers (3,2,7) are the quotients. In general, given any two numbers \(a,b\), they can be written as \[a=b\cdot q +r\]where q=quotient and r=remainder.

  5. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So the point is that we must repeat it again and again till we get the remainder 0? That does make sense.

  6. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    May I have a practice problem?

  7. KingGeorge
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 6

    Try it out on \(a=342\) and \(b=295\).

  8. KingGeorge
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 6

    One more thing, make sure \(0\leq r<b\) in the equation \(a=b\cdot q+r\).

  9. asnaseer
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @KingGeorge - wonderful explanation! :)

  10. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I wish that the Chrome Aw Snap didn't exist.

  11. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    OMG I never knew it's called euclidean algorithm. I love this method. lol

  12. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[342= 295\cdot 1 + 47 \]\[295 = 47 \cdot 6+13 \]\[47 = 13\cdot 3+8 \]\[13 = 8\cdot 1 + 5 \]\[8 = 5 \cdot 1 + 3 \]\[5 = 3 \cdot 1 + 2 \]\[ 3 = 2 \cdot 1 + 1\]\[2 = 1 \cdot 2 + 0 \]

  13. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So, they are co-prime!

  14. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Thank you :)

  15. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Let me return back to that CRT question. Okay.

  16. KingGeorge
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 6

    You're welcome. If you want a proof of the algorithm, I could probably type that up as well.

  17. ParthKohli
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Haha no :P

  18. Not the answer you are looking for?
    Search for more explanations.

    Search OpenStudy
    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.