Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics IV / 7 / a / ii

Back-substitution in the Euclidean algorithm gives the Bezout identityMultiplying by five produces a particular solution $(x,y)=(-115,60)$. Since $259/7=37$ and $497/7=71$ are coprime integers, all solutions of the linear Diophantine equation areReplacing $t$ by $t+2$ gives the smaller equivalent starting solutionIndeed $259\cdot 27-497\cdot 14=35$. Conversely, subtracting this solution from any other gives $37(x-27)=-71(y+14)$. Coprimality forces $71$ to divide $x-27$, and then $y+14$ is the corresponding multiple of $-37$. This proves completeness of the family.