Question 155
Let $n = p_{1} p_{2} \ldots p_{2017}$ be the positive integer where $p_{1}, p_{2}, \ldots, p_{2017}$ are 2017 distinct odd primes. A triangle is called nice if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are considered different if their tuples of side lengths are different).