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

KingGeorgeBest ResponseYou've already chosen the best response.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, \(\gcd(572,165)=11.\)
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.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.
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.1
So the point is that we must repeat it again and again till we get the remainder 0? That does make sense.
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.1
May I have a practice problem?
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.6
Try it out on \(a=342\) and \(b=295\).
 one year ago

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

asnaseerBest ResponseYou've already chosen the best response.0
@KingGeorge  wonderful explanation! :)
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.1
I wish that the Chrome Aw Snap didn't exist.
 one year ago

Ishaan94Best ResponseYou've already chosen the best response.0
OMG I never knew it's called euclidean algorithm. I love this method. lol
 one year ago

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

ParthKohliBest ResponseYou've already chosen the best response.1
So, they are coprime!
 one year ago

ParthKohliBest ResponseYou've already chosen the best response.1
Let me return back to that CRT question. Okay.
 one year ago

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