Solving a linear Diophantine equation

EasyBrainteasers~4m

Find all integer pairs $(x, y)$ satisfying $3x + 7y = 10000$ . Every solution can be written as $x = a + 7n$ and $y = b - 3n$ for an arbitrary integer $n$ , where $a, b$ are the integers obtained when $b$ is taken to be its smallest positive value. What is $a \cdot b$ ?

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Reduce the equation modulo 3 to isolate a condition on $y$ alone.
You get $y \equiv 1 \pmod 3$ ; take the smallest positive $y$ as $b$ , then solve for $x = a$ .

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Asked at: Jane Street, Citadel

Solving a linear Diophantine equation

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