How to prove the theorem stated [here](https://math.stackexchange.com/questions/20737/prime-powers-that-divide-a-factorial).

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> **Theorem.** (Kummer, 1854) *Let $p$ be a prime. The highest power of $p$ that divides the binomial coefficient $\binom{m+n}{n}$ is equal to the number of "carries" when adding $m$ and $n$ in base $p$.*
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So far, I know if $m+n$ can be expanded in base power as 
$$m+n= a\_0 + a\_1 p + \dots +a\_k p^k$$
and $m$ have to coefficients $\{ b\_0 , b\_1 , \dots b\_i\}$ and $n$ can be expanded with coefficients $\{c\_0, c\_1 ,\dots , c\_j\}$ in base $p$ then the highest power of prime that divides $\binom{m+n}{n}$ can be expressed as 
$$e = \frac{(b\_0 + b\_1 + \dots b\_i )+ (c\_0 + c\_1 + \dots c\_j )-(a\_0 + a\_1 + \dots a\_k )}{p-1}$$
It follows from [here](http://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-spring-2012/lecture-notes/MIT18_781S12_lec3.pdf) page number $4$. But how does it relate to the number of carries? I am not being able to connect. Perhaps I am not understanding something very fundamental about addition.