Lucas' theorem

In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ${\displaystyle {\tbinom {m}{n}}}$ by a prime number p in terms of the base p expansions of the integers m and n.

Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.

For non-negative integers m and n and a prime p, the following congruence relation holds:

where

and

are the base p expansions of m and n respectively. This uses the convention that ${\displaystyle {\tbinom {m}{n}}=0}$ if m < n.

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