KMO 2016

Let $N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}$ . Prove that there are $(b_1+1)(b_2+1)\ldots(b_k+1)$ number of $n$ s which satisfies these two conditions. $\frac{n(n+1)}{2}\le N$ , $N-\frac{n(n+1)}{2}$ is divided by $n$ .

A subset $S \in \{0, 1, 2, \cdots , 2000\}$ satisfies $|S|=401$ .

Prove that there exists a positive integer $n$ such that there are at least $70$ positive integers $x$ such that $x, x+n \in S$