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If $a,b$ are nonzero integers, $d=\gcd (a,b)$, and $(x_0,y_0)$ is one solution of the linear Diophantine equation $ax+by=c$, then all solutions areSubtract two solutions and use the fact that $a/d$ and $b/d$ are coprime integers. The linear Diophantine equation solvability criterion says that a particular solution exists exactly when $d$ divides $c$; the extended Euclidean algorithm constructs one. This parametrization separates existence from the complete family of answers.

Applying the Euclidean algorithm,Hence $\gcd (2508,3211)=19$. Since $19$ does not divide $55$, the linear Diophantine equation solvability criterion gives

The linear Diophantine equation solvability criterion says that $ax+by=c$ has an integer solution exactly when $\gcd (a,b)$ divides $c$. The Euclidean algorithm givesThus $\gcd (1007,2314)=1$, which divides $37$. Therefore