Kummer's theorem is a result in number theory that relates to the divisibility of binomial coefficients. Specifically, it provides a criterion for determining when a binomial coefficient, denoted as n \choose k, is divisible by a prime number p. The theorem states that this divisibility can be analyzed using the base p representations of the integers n and k. The theorem is named after the mathematician Ernst Eduard Kummer, who developed it in the 19th century. Kummer's theorem has applications in various areas of mathematics, including combinatorics and algebra, and it helps in understanding the structure of p-adic numbers and their properties.