Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics IV / 7 / a / Solution

The greatest common divisor of $54$ and $21$ is $3$, which divides $906$. Divide the linear Diophantine equation by $3$ to obtainReducing modulo $7$ gives $4x\equiv 1(\mathrm{mod}7)$, so $x\equiv 2(\mathrm{mod}7)$. Substitution gives the complete family of integer solutionsEvery 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 $0\le t\le 2$. Thus the pairs with both entries positive are