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.

Answer 2

There are six possible remainders modulo 6: those are $0, 1, 2, 3, 4, 5$.

Since there are $7$ integers in your set, by the pigeonhole principle there will be at least two of them with the same remainder modulo 6. Their difference will then be divisible by $6$.

Answer 3

By the remainder's theorem, every integer number divided by $6$ is of the form: $$n=6q+r$$ where $0 \leq r < 6$

Since there are $6$ possible remainders, while you have $7$ numbers, at least two numbers must have the same remainder divided by $6$. Therefore, the difference of these two numbers must be divisible by $6$.