IR-TST 2023 Problem 7

Suppose that $d(n)$ is the number of positive divisors of natural number $n$ . Prove that there is a natural number $n$ such that $$\forall i\in \mathbb{N} , i \le 1402: \frac{d(n)}{d(n \pm i)} >1401$$ Proposed by Navid Safaei and Mohammadamin Sharifi