Problem: Let $a$, $b$ be integers greater than $2$.
Prove that there exists a positive integer $k$
and a finite sequence $n_1, n_2, \dots, n_k$ of
positive integers such that $n_1 = a$, $n_k = b$,
and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$
for each $i$ ($1 \leq i < k$).

Solution: Consider a graph $G$ on the vertex set $\{3, 4, \dots\}$
and with edges between $v$, $w$ if $v + w \mid vw$;
the problem is equivalent to showing that $G$ is connected.

First, note that $n$ is connected to $n(n-1)$, $n(n-1)(n-2)$, etc.\
up to $n!$.
But for $n > 2$, $n!$ is connected to $(n+1)!$ too:
\begin{itemize}
  \ii $n! \to (n+1)!$ if $n$ is even
  \ii $n! \to 2n! \to (n+1)!$ if $n$ is odd.
\end{itemize}
This concludes the problem.