Kummer's theorem

In mathematics, Kummer's theorem for binomial coefficients gives the p-adic valuation of a binomial coefficient, i.e., the exponent of the highest power of a prime number p dividing this binomial coefficient. The theorem is named after Ernst Kummer, who proved it in the paper Kummer (1852).

Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation ${\displaystyle \nu _{p}\left({\tbinom {n}{m}}\right)}$ is equal to the number of carries when m is added to n − m in base p.

It can be proved by writing ${\displaystyle {\tbinom {n}{m}}}$ as ${\displaystyle {\tfrac {n!}{m!(n-m)!}}}$ and using Legendre's formula.

...
Wikipedia