Product of Even Numbers

For there to be only one way to express $N$ as a product of $p$ and $q$ through the conditions, $N$ must be the product of $4$ and a prime, $k$ .

This means that $N = 2^2 * k$ .

There is only one pair of integers $p$ and $q$ that will suffice here, and they are $2$ and $2k$ .

$k$ must be the largest prime $≤\left \lfloor{\frac{2017}{4}}\right \rfloor =504$ .

Thus $k$ is 503, and $N$ is $4k = 4*503 = 2012$ .