Achieving 1/lcm

Let $d_n$ be the least common multiple of $1,2,\dots, n$. For eg, $d_4 = 12, d_5 = 60, d_9 = 2520$.

Given an $n \ge 1$, Show that there exist integers $A_1, A_2, \dots A_n$ (not necessarily positive) such that

$$\sum_{k=1}^{n} \dfrac{A_k}{k} = \dfrac{1}{d_n}$$



[Solution]