Kummer's theorem

Statement

Let $p$ be a prime number, and let $m$ and $n$ be natural numbers. Then $\nu_p \binom{m + n}{m}$ is the number of carries when one adds $m$ and $n$ in base $p$. Here $\nu_p$ is the p-adic valuation.

Remarks

For example, in base $2$, $3 + 1 = 11_2 + 1_2 = 100_2$ has $2$ carries, and $\binom{4}{1} = 4 = 2^2$. The theorem follows from Legendre's formula in the form with the digit sums $s_p$. A consequence: $p \mid \binom{m+n}{m}$ if and only if there is at least one carry.

History

Ernst Kummer proved the theorem in 1852.

Explanations

L1 human

Write $m$ and $n$ in base $p$ and add them by hand. Count how many times you carry a $1$. That count is the exponent of $p$ in the binomial coefficient $\binom{m+n}{m}$. For $p = 10$ the rule fails, because $10$ is not prime. The theorem needs a prime base.