[Math] Proving a statement using Pigeonhole principle

Question

I am trying to understand how to prove a Statement using the pigeonhole principle.

Prove the following result using the pigeon-hole principle.
In every collection of 7 integers there are at least two whose difference is divisible by 6.
any ideas? thanks in advance

Answer 1

Consider the residues of the 7 numbers when divided by 6 (their classes modulo 6). There are 6 possible residues, so one must be used twice, by the pigeonhole principle. The difference of the corresponding original integers is a multiple of 6.