Lucas' theorem

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In number theory, the Lucas' theorem states the following: Let m and n be non-negative integers and p a prime. Let

m=m_kp^k+m_{k-1}p^{k-1}+\cdots +m_1p+m_0, and
n=n_kp^k+n_{k-1}p^{k-1}+\cdots +n_1p+n_0

be the base p expansions of m and n respectively. Then

\binom{m}{n}\equiv\prod_{i=0}^k\binom{m_i}{n_i}\pmod p.


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