Maths problems

Could you give me some solution for those questions?

1. prove 'n^3 + 5n is divisible by 6' without using induction

2. let m, n be coprime positive integers, with m is greater than 1.
Prove that if log(base m)n is rational then it is 0.

3. there are six towns, such that between each pair of towns there is either a train or bus service (but not both). Prove that there are three towns that can be visited in a loop (going via no other towns) using only on mode of transport.

3. there are six towns, such that between each pair of towns there is either a train or bus service (but not both). Prove that there are three towns that can be visited in a loop (going via no other towns) using only on mode of transport.

if i am not wrong the 3rd question involves the pigeonhole principle, if ur trying qns 3 u gotta be patient becuase it will involves a lot of reasoning.

hmm using a graph theoretical approach?

Well, there Is a modified version of the pigeonhole principle involved. If there are 5 pigeons that go into 2 holes, 1 hole will receive at least $\lceil \frac{5}{2} \rceil=3$ pigeons. If this does not make sense, try to fill those holes while keeping all of them with less than 3 pigeons.