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ParthKohli
 3 years ago
Please explain me the Euclidean Algorithm.
Help appreciated.
ParthKohli
 3 years ago
Please explain me the Euclidean Algorithm. Help appreciated.

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KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.6First step, look at an example. I'll color things to make it easy to see what I'm doing.

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.6Look 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, \(\gcd(572,165)=11.\)

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.6The 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
 3 years ago
Best ResponseYou've already chosen the best response.1So the point is that we must repeat it again and again till we get the remainder 0? That does make sense.

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.1May I have a practice problem?

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.6Try it out on \(a=342\) and \(b=295\).

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.6One more thing, make sure \(0\leq r<b\) in the equation \(a=b\cdot q+r\).

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.0@KingGeorge  wonderful explanation! :)

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.1I wish that the Chrome Aw Snap didn't exist.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0OMG I never knew it's called euclidean algorithm. I love this method. lol

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.1\[342= 295\cdot 1 + 47 \]\[295 = 47 \cdot 6+13 \]\[47 = 13\cdot 3+8 \]\[13 = 8\cdot 1 + 5 \]\[8 = 5 \cdot 1 + 3 \]\[5 = 3 \cdot 1 + 2 \]\[ 3 = 2 \cdot 1 + 1\]\[2 = 1 \cdot 2 + 0 \]

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.1So, they are coprime!

ParthKohli
 3 years ago
Best ResponseYou've already chosen the best response.1Let me return back to that CRT question. Okay.

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.6You're welcome. If you want a proof of the algorithm, I could probably type that up as well.
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