Question 143
Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b \gt n$ such that
$$
\prod_{i=1}^{2015}(a+i) \mid b(b+2016) ; \quad \prod_{i=1}^{2015}(a+i) \nmid b ; \quad \prod_{i=1}^{2015}(a+i) \nmid(b+2016) .
$$
$$
\prod_{i=1}^{2015}(a+i) \mid b(b+2016) ; \quad \prod_{i=1}^{2015}(a+i) \nmid b ; \quad \prod_{i=1}^{2015}(a+i) \nmid(b+2016) .
$$