Use Euclidean Algorithm to find $gcd(42823,6409) $ and find the values of integer $x$ and $y$ satisfying $gcd(42823,6409) = 42823 \cdot x + 6409 \cdot y$.

Example: Use Euclidean Algorithm to find $gcd(42823,6409) $ and find the values of integer $x$ and $y$ satisfying $gcd(42823,6409) = 42823 \cdot x + 6409 \cdot y$.

Answer: To find  $gcd(42823,6409) $ using Euclidean Algorithm. We have to apply Division Algorithm to integers $42823$ and $6409$. Firstly we divide $42823$ by $6409$. When we divide $42823$ by $6409$, we get $q = 6$ (i.e. $6$ is a quotient) and $r = 4369$ (i.e. $4369$ is a remainder). So we get following system of equations in  Euclidean Algorithm.

$42823 = 6409~ \cdot ~6 + 4369$

$6409 = 4369 ~\cdot ~1 + 2040$

$4369 = 2040 ~\cdot ~2 + 289$

$2040 = 289 ~\cdot~ 7 + 17$

$289 = 17 ~\cdot ~17 + 0$

$17$ is the last nonzero remainder obtained in this Algorithm, so  $gcd(42823,6409)  = 17$.


From the above equations, putting value of remainder (eliminating remainder i.e. $17, ~289,~2040,~ 4369$) and simplifying we can find value of integers $x$ and $y$ such that $gcd(42823,6409) = 42823 \cdot x + 6409 \cdot y$.

We can write $gcd(42823,6409) = 17$ as follows

$17 = 2040 ~-~ 289 ~\cdot~ 7$

$17 = 2040 ~-~ (4369 ~-~ 2040 ~\cdot~ 2  ) ~\cdot~ 7$

$17 = 2040 ~-~ 4369 ~\cdot~ 7 + 2040 ~\cdot~ 14$

$17 = 2040 ~\cdot~ 15 ~-~ 4369 ~\cdot~ 7  $

$17 = (6409 ~-~ 4369 ~\cdot~ 1 )~\cdot~ 15 ~-~ 4369 ~\cdot~ 7  $

$17 = 6409 ~\cdot~ 15~ -~ 4369 ~\cdot~ 15  - 4369 ~\cdot~ 7  $

$17 = 6409 ~\cdot~ 15 ~-~ 4369 ~\cdot~ 22  $

$17 = 6409 ~\cdot~ 15~ -~ (42823 ~-~ 6409 ~\cdot~ 6 ) ~\cdot~ 22  $

$17 = 6409 ~\cdot~ 15 ~-~ 42823 ~\cdot~ 22 + 6409 ~\cdot~ 132   $

$17 = 6409 ~\cdot~ 147 ~-~ 42823 ~\cdot~ 22   $

$17 = 6409 ~\cdot~ 147 + 42823 ~\cdot~ ( - 22)   $

$17 =42823 ~\cdot~ ( - 22)  +  6409 ~\cdot~ 147  $

comparing with $gcd(42823,6409) = 17 = 42823 \cdot x + 6409 \cdot y$, we get $x = - 22 $ and $y = 147  $.

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