# example of pigeonhole principle

A 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 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 $7$. ∎

Title example of pigeonhole principle ExampleOfPigeonholePrinciple 2013-03-22 12:41:32 2013-03-22 12:41:32 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Example msc 05-00