# Lucas' theorem Consequence

Lucas' theorem consequence

$$\binom {m}{n}=\ \prod_{i=0}^k\;\ \binom {m_i}{n_i} \pmod{p},$$ $$m=m_k\;p^k+m_{k-1}\;p^{k-1}+\cdots +m_1\; p+m_0,$$ $$n=n_k\;p^k+n_{k-1}\;p^{k-1}+\cdots +n_1\;p+n_0$$

Lucas' theorem states that a binomial coefficient $C(m,n)$ is divisible by a prime $p$ if and only if at least one of the base $p$ digits of $n$ is greater than the corresponding digit of $m$.

Assuming $n_i$ is greater than $m_i$, then what is $C(m_i,n_i)$?

Can someone explain this to me? Thank you. I can't post images.

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The binomial coefficient is zero if its lower index is bigger than its upper index. –  Ｊ. Ｍ. Aug 1 '12 at 8:04
Your images of formulas are unreadable to me. Cannot you put them here using this site Latex engine? –  enzotib Aug 1 '12 at 8:24