Question 79
For any positive integer $n$, let $N=\varphi(1)+\varphi(2)+\ldots+\varphi(n)$. Show that there exists a sequence
$$
a_{1}, a_{2}, \ldots, a_{N}
$$
containing exactly $\varphi(k)$ instances of $k$ for all positive integers $k \leq n$ such that
$$
\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\cdots+\frac{1}{a_{N} a_{1}}=1 .
$$
$$
a_{1}, a_{2}, \ldots, a_{N}
$$
containing exactly $\varphi(k)$ instances of $k$ for all positive integers $k \leq n$ such that
$$
\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\cdots+\frac{1}{a_{N} a_{1}}=1 .
$$