Prove that among eight integers, there are always two whose difference is divisible by seven.
Let's write each number as .
If for some , then the difference between the th and th number
which is divisible by .
Now we prove that there must always be some such that via the Pigeonhole Principle.
The possible set of values for is .
The set only contains values while we have integers. So by the pigeonhole principle, one of the values will be repeated.