Simple Addition Question

Question

In basic math, when adding there is the notion of carry. For example if you have $9999+9999$, since $9+9=18$ the $1$ is leftover and carried to the top. Is there some formula that gives us the total number carried? So in the case of our example, it would be $4$.

Answer 1

Kummer's theorem says the number of carries when adding $m + n$ in a prime base $p$ is the largest integer $k$ such that $p^k$ divides the binomial coefficient ${m + n \choose m}$. Unfortunately, $10$ is not a prime.