lucas 定理简单证明

已知 \(p\) 是质数。

\[\because x^p\equiv x\pmod p \]

\[\text{令 }p\text{ 进制下 }a=\sum a_ip^i，b=\sum b_ip^i \]

\[\therefore (1+x)^a\equiv (1+x)^{\sum a_ip^i}\equiv (1+x)^{\sum a_i}\equiv\prod(1+x^{p^i})^{a_i}\pmod p \]

\[\therefore\prod\left[\sum\binom{a_i}{b_i}x^{b_ip^i}\right]\equiv\sum\left[\prod\binom{a_i}{b_i}\right]x^b\equiv\sum\binom{\sum a_ip^i}{b}x^b\pmod p \]

\[\prod\binom{a_i}{b_i}\equiv\binom{a}{b}\pmod p \]

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