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ParthKohli Group Title

Please explain me the Euclidean Algorithm. Help appreciated.

  • 2 years ago
  • 2 years ago

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

    • 2 years ago
  2. ParthKohli Group Title
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    Okay - I follow.

    • 2 years ago
  3. KingGeorge Group Title
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    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, \(\gcd(572,165)=11.\)

    • 2 years ago
  4. KingGeorge Group Title
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    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.

    • 2 years ago
  5. ParthKohli Group Title
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    So the point is that we must repeat it again and again till we get the remainder 0? That does make sense.

    • 2 years ago
  6. ParthKohli Group Title
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    May I have a practice problem?

    • 2 years ago
  7. KingGeorge Group Title
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    Try it out on \(a=342\) and \(b=295\).

    • 2 years ago
  8. KingGeorge Group Title
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    One more thing, make sure \(0\leq r<b\) in the equation \(a=b\cdot q+r\).

    • 2 years ago
  9. asnaseer Group Title
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    @KingGeorge - wonderful explanation! :)

    • 2 years ago
  10. ParthKohli Group Title
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    I wish that the Chrome Aw Snap didn't exist.

    • 2 years ago
  11. Ishaan94 Group Title
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    OMG I never knew it's called euclidean algorithm. I love this method. lol

    • 2 years ago
  12. ParthKohli Group Title
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    \[342= 295\cdot 1 + 47 \]\[295 = 47 \cdot 6+13 \]\[47 = 13\cdot 3+8 \]\[13 = 8\cdot 1 + 5 \]\[8 = 5 \cdot 1 + 3 \]\[5 = 3 \cdot 1 + 2 \]\[ 3 = 2 \cdot 1 + 1\]\[2 = 1 \cdot 2 + 0 \]

    • 2 years ago
  13. ParthKohli Group Title
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    So, they are co-prime!

    • 2 years ago
  14. ParthKohli Group Title
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    Thank you :)

    • 2 years ago
  15. ParthKohli Group Title
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    Let me return back to that CRT question. Okay.

    • 2 years ago
  16. KingGeorge Group Title
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    You're welcome. If you want a proof of the algorithm, I could probably type that up as well.

    • 2 years ago
  17. ParthKohli Group Title
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    Haha no :P

    • 2 years ago
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