Question 73
Angela has the polynomials $x-1$, $(x-1)(x-2)$, $(x-1)(x-2)(x-3)$, $\ldots$ up to $(x-1)(x-2) \cdots (x-2017)(x-2018)$ at her disposal, and she divides them into two groups. Letting $p(x)$ be the product of the polynomials of the first group and $q(x)$ that of the polynomials of the second group, Angela notices that the polynomial $p(x)$ divides the polynomial $q(x)$, and that the degree of the quotient $\frac{q(x)}{p(x)}$ is the smallest possible: what is this degree?