General integer solution of 3x + 7y = 10000

Every integer pair $(x,y)$ with $3x+7y=10000$ can be written as $x=a+7n$, $y=b-3n$ for $n\in\mathbb{Z}$, where the constants $a,b$ are integers chosen so that $b$ is the smallest possible positive value. Compute $ab$.

$3331$. Since $10000\equiv1\pmod 3$, $7y\equiv1\pmod3$ forces $y\equiv1\pmod 3$, so the smallest positive $y$ is $b=1$, with $x=(10000-7)/3=3331$. Hence $a=3331$, $b=1$ and $ab=3331$.