example of pigeonhole principle

A example.

Theorem.

For any set of $\mathrm{8}$ integers, there exist at least two of themwhose difference is divisible by $\mathrm{7}$.

Proof.

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