= Solution
{c}
The <greatest common divisor> of $54$ and $21$ is $3$, which divides $906$. Divide the <linear Diophantine equation> by $3$ to obtain
$$
18x+7y=302.
$$
Reducing modulo $7$ gives $4x\equiv1\pmod7$, so $x\equiv2\pmod7$. Substitution gives the complete family of <integer> solutions
$$
\boxed{x=2+7t,\qquad y=38-18t,\qquad t\in\mathbb Z.}
$$
Every such pair satisfies the <equation>, and every solution must have this residue for $x$, so none are omitted. Positivity requires $2+7t>0$ and $38-18t>0$. For <integer> $t$ these are exactly $0\leq t\leq2$. Thus the pairs with both entries positive are
$$
\boxed{(x,y)=(2,38),\ (9,20),\ (16,2).}
$$
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