TUYMAADA 2014 J8

There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$ . Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong.

(K. Kokhas)