ParthKohli
  • ParthKohli
Please explain me the Euclidean Algorithm. Help appreciated.
Mathematics
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SOLVED
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chestercat
  • chestercat
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KingGeorge
  • KingGeorge
First step, look at an example. I'll color things to make it easy to see what I'm doing.
ParthKohli
  • ParthKohli
Okay - I follow.
KingGeorge
  • KingGeorge
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, \(\gcd(572,165)=11.\)

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KingGeorge
  • KingGeorge
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.
ParthKohli
  • ParthKohli
So the point is that we must repeat it again and again till we get the remainder 0? That does make sense.
ParthKohli
  • ParthKohli
May I have a practice problem?
KingGeorge
  • KingGeorge
Try it out on \(a=342\) and \(b=295\).
KingGeorge
  • KingGeorge
One more thing, make sure \(0\leq r
asnaseer
  • asnaseer
@KingGeorge - wonderful explanation! :)
ParthKohli
  • ParthKohli
I wish that the Chrome Aw Snap didn't exist.
anonymous
  • anonymous
OMG I never knew it's called euclidean algorithm. I love this method. lol
ParthKohli
  • ParthKohli
\[342= 295\cdot 1 + 47 \]\[295 = 47 \cdot 6+13 \]\[47 = 13\cdot 3+8 \]\[13 = 8\cdot 1 + 5 \]\[8 = 5 \cdot 1 + 3 \]\[5 = 3 \cdot 1 + 2 \]\[ 3 = 2 \cdot 1 + 1\]\[2 = 1 \cdot 2 + 0 \]
ParthKohli
  • ParthKohli
So, they are co-prime!
ParthKohli
  • ParthKohli
Thank you :)
ParthKohli
  • ParthKohli
Let me return back to that CRT question. Okay.
KingGeorge
  • KingGeorge
You're welcome. If you want a proof of the algorithm, I could probably type that up as well.
ParthKohli
  • ParthKohli
Haha no :P

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