Question 186
Let $m \ge 2$ and $n \ge 3$ be natural numbers. Prove that there exist $m$ distinct natural numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{m}$, all divisible by $n-1$, such that
$$
\frac{1}{n}=\frac{1}{a_{1}}-\frac{1}{a_{2}}+\frac{1}{a_{3}}-\ldots+(-1)^{m-1} \frac{1}{a_{m}}.
$$
$$
\frac{1}{n}=\frac{1}{a_{1}}-\frac{1}{a_{2}}+\frac{1}{a_{3}}-\ldots+(-1)^{m-1} \frac{1}{a_{m}}.
$$