Problem

Source: CWMO 2003, Problem 3

Tags: modular arithmetic, pigeonhole principle, number theory unsolved, number theory



Let n be a given positive integer. Find the smallest positive integer un such that for any positive integer d, in any un consecutive odd positive integers, the number of them that can be divided by d is not smaller than the number of odd integers among 1,3,5,,2n1 that can be divided by d.