Let's say we have binomial coefficient $\binom{n}{m}$. And we need to find the greatest power of prime $p$ that divides it.

Usually Kummer's theorem is stated in terms of the number of carries you perform while adding $m$ and $n-m$ in base $p$.

I found an equivalent statement of this theorem that reads like this: if we write $$ \binom{n}{m}\equiv\binom{n_0}{m_0}\binom{n_1}{m_1}\ldots\binom{n_d}{m_d}\pmod{p}, $$ where $n = n_0 + n_1p + n_2p^2 + \ldots + n_dp^d$ and $m = m_0 + m_1p + m_2p^2 + \ldots + m_dp^d$, then the power dividing $\binom{n}{m}$ is precisely the number of indicies $i$ for which $n_i<m_i$.

Now let's take an example. Let's look at $\binom{25}{1}$ and $p=5$. We have $$ \binom{25}{1}\equiv\binom{1}{0}\binom{0}{0}\binom{0}{1}\pmod{5}. $$ We have only one index $i$ for which $n_i < m_i$, which is the last one. This suggests that $\binom{25}{1}$ can't be divided by $25$, which obviously isn't true.

Where's the problem? In case you wonder where I found this statement of Kummer's theorem, here is the link: http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf

Thank you!