--description--

For integers $a$ and $b$, we define $D(a, b)$ as the domain enclosed by the parabola $y = x^2$ and the line $y = ax + b: D(a, b) = \{ (x, y) | x^2 ≤ y ≤ ax + b \}$.

$L(a, b)$ is defined as the number of lattice points contained in $D(a, b)$. For example, $L(1, 2) = 8$ and $L(2, -1) = 1$.

We also define $S(N)$ as the sum of $L(a, b)$ for all the pairs ($a$, $b$) such that the area of $D(a, b)$ is a rational number and $|a|,|b| ≤ N$.

We can verify that $S(5) = 344$ and $S(100) = 26\,709\,528$.

Find $S({10}^{12})$. Give your answer $\bmod {10}^8$.

--hints--

latticePoints() should return 18224771.

assert.strictEqual(latticePoints(), 18224771);

--seed--

--seed-contents--

function latticePoints() {

  return true;
}

latticePoints();

--solutions--

// solution required