Kummer's Theorem

Theorem
Let $p$ be a prime number.

Let $a, b \in \Z_{\ge 0}$.

Let:
 * $p^n \divides \dbinom {a + b} b$

but
 * $p^{n + 1} \nmid \dbinom {a + b} b$

where:
 * $\divides$ denotes divisibility
 * $\nmid$ denotes non-divisibility
 * $\dbinom {a + b} b$ denotes a binomial coefficient.

Then $n$ equals the number of carries that occur when $a$ is added to $b$ using the classical addition algorithm in base $p$.