Let $m, n,$ and $k$ be natural numbers, where $n$ is odd. Prove that $\frac{1}{m}+\frac{1}{m+n}+...+\frac{1}{m+kn}$ is not a natural number.
Problem
Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade
Tags: number theory, modulo