I just came across this cute little result on Quora and generalised it to the following.

**Proposition. **For any integers ,

divides .

*First proof. *Note that

.

Since , the result follows.

*Second proof. *Let and where are the prime factors of . For each , the base representations of and have and trailing zeros respectively. Hence by Kummer’s theorem divides . Hence

divides . Now the result follows using .

A nice corollary is the following property of Pascal’s triangle.

**Corollary. **For any integers ,

.

Using the identity

the argument in the first proof above can be adapted to prove the following generalisation:

**Proposition. **For any integers ,

divides .

**Corollary. **Any two entries in a given row of Pascal’s triangle have a common factor .