= Solution
{c}
Back-substitute in the <Euclidean algorithm> to obtain an <integer linear combination>:
$$
\begin{aligned}
9&=45-36=45-(306-6\cdot45)\\
&=7\cdot45-306=7(657-2\cdot306)-306\\
&=7\cdot657-15\cdot306.
\end{aligned}
$$
Multiplying by $3$ gives the particular solution $(x_0,y_0)=(-45,21)$ of the <linear Diophantine equation>. Subtracting it from any other solution and dividing by $9$ gives
$$
34(x+45)+73(y-21)=0.
$$
The <integers> $34$ and $73$ are <coprime>, so $73$ divides $x+45$. Writing $x+45=73t$ forces $y-21=-34t$. Conversely these expressions satisfy the original equation for every <integer> $t$. Thus <all solutions of a linear Diophantine equation> here are
$$
\boxed{(x,y)=(-45+73t,\ 21-34t),\qquad t\in\mathbb Z.}
$$
Back to article page
