Kummer’s Theorm

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Kummer’s theorm gives the exponent of the highest power of a prime number p dividing this binomial coefficient.

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

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