Determine all solutions in the integers of Diophantine equation $56x + 72y = 40$.

 Example: Determine all solutions in the integers of the Diophantine equation $56x + 72y = 40$.

Proof:  Firstly, we will find $gcd(56,72)$ using Euclidean Algorithm. We use Euclidean Algorithm and we find that

$72 = 56 ~\cdot~ 1 + 16$

$56 = 16 ~\cdot~ 3 + 8$

$16 = 8 ~\cdot~ 2 + 0$

$8$ is the least nonzero remainder obtained in this Algorithm, so $gcd(56,72) = 8$.

Since $8 ~ | ~ 40$,  a solution to this equation exists. To obtain the integer $8$ as a linear combination of $56$ and $72$, we work backward through the previous calculations, as follows: 

$8 = 56 - 16 ~\cdot~ 3$

$8 = 56 - (72 - 56 ~\cdot~ 1) ~\cdot~ 3$

$8 = 56 - 72 ~\cdot~ 3 + 56 ~\cdot~ 3$

$8 = 56 ~\cdot~ 4 - 72 ~\cdot~ 3$

$8 = 56 ~\cdot~ 4 + 72 ~\cdot~ (-3)$

We multiply both side of above equation by $5$. Hence we will get 

$40 = 56 ~\cdot~ 20 + 72 ~\cdot~ (-15)$

so that $x = 20$ and $ y = - 15$ is a one solution of given Diophantine equation. Other solutions are given by 

$x = 20 + (\frac{72}{8} ) \cdot t = 20 + 9 \cdot t$

$y = - 15 + (\frac{56}{8} ) \cdot t = - 15 + 7 \cdot t $, where $t$ is an arbitrary integer.

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