Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most $n$.
Problem
Source: 2017 Romania JBMO TST 4.2 - Kurschak Competition, 1999
Tags: inequalities, number theory, Divisors, Difference