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 is equal to the number of carries when m is added to n − m in base p.
It can be proved by writing as and using Legendre's formula.
Multinomial coefficient generalization
Kummer's theorem may be generalized to multinomial coefficients as follows: Write the base- expansion of an integer as , and define to be the sum of the base- digits. Then