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# example of pigeonhole principle

A simple example.

###### Theorem.

For any set of $8$ integers, there exist at least two of them whose difference is divisible by $7$.

###### Proof.

The residue classes modulo $7$ are $0,1,2,3,4,5,6$. We have seven classes and eight integers. So it must be the case that 2 integers fall on the same residue class, and therefore their difference will be divisible by $7$. ∎

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