= Solution
{c}
Back-substitution in the <Euclidean algorithm> gives the <Bezout identity>
$$
7=238-11\cdot21=12\cdot238-11\cdot259=12\cdot497-23\cdot259.
$$
Multiplying 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> are
$$
(x,y)=(-115+71t,\ 60-37t),\qquad t\in\mathbb Z.
$$
Replacing $t$ by $t+2$ gives the smaller equivalent starting solution
$$
\boxed{(x,y)=(27+71t,\ -14-37t),\qquad t\in\mathbb Z.}
$$
Indeed $259\cdot27-497\cdot14=35$. 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.
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