IR-TST 2012 Problem 7

Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$ ), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$ ?

Proposed by Morteza Saghafian